Metamath Proof Explorer Home Metamath Proof Explorer
Most Recent Proofs 		 
Mirrors  >  Home  >  MPE Home  >  Th. List  >  Recent ILE Most Recent             Other  >  MM 100
Most recent proofs    These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the set.mm database for the Metamath Proof Explorer (and the Hilbert Space Explorer). The set.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from develop commit 212ee7d9, also available here: set.mm (43MB) or set.mm.bz2 (compressed, 13MB).

The original proofs of theorems with recently shortened proofs can often be found by appending "OLD" to the theorem name, for example 19.43OLD for 19.43. The "OLD" versions are usually deleted after a year.

Other links    Email: Norm Megill.    Mailing list: Metamath Google Group Updated 7-Dec-2021 .    Contributing: How can I contribute to Metamath?    Syndication: RSS feed (courtesy of Dan Getz)    Related wikis: Ghilbert site; Ghilbert Google Group.

Recent news items    (7-Aug-2021) Version 0.198 of the metamath program fixes a bug in "write source ... /rewrap" that prevented end-of-sentence punctuation from appearing in column 79, causing some rewrapped lines to be shorter than necessary. Because this affects about 2000 lines in set.mm, you should use version 0.198 or later for rewrapping before submitting to GitHub.

(7-May-2021) Mario Carneiro has written a Metamath verifier in Lean.

(5-May-2021) Marnix Klooster has written a Metamath verifier in Zig.

(24-Mar-2021) Metamath was mentioned in a couple of articles about OpenAI: Researchers find that large language models struggle with math and What Is GPT-F?.

(26-Dec-2020) Version 0.194 of the metamath program adds the keyword "htmlexturl" to the $t comment to specify external versions of theorem pages. This keyward has been added to set.mm, and you must update your local copy of set.mm for "verify markup" to pass with the new program version.

(19-Dec-2020) Aleksandr A. Adamov has translated the Wikipedia Metamath page into Russian.

(19-Nov-2020) Eric Schmidt's checkmm.cpp was used as a test case for C'est, "a non-standard version of the C++20 standard library, with enhanced support for compile-time evaluation." See C++20 Compile-time Metamath Proof Verification using C'est.

(10-Nov-2020) Filip Cernatescu has updated the XPuzzle (Android app) to version 1.2. XPuzzle is a puzzle with math formulas derived from the Metamath system. At the bottom of the web page is a link to the Google Play Store, where the app can be found.

(7-Nov-2020) Richard Penner created a cross-reference guide between Frege's logic notation and the notation used by set.mm.

(4-Sep-2020) Version 0.192 of the metamath program adds the qualifier '/extract' to 'write source'. See 'help write source' and also this Google Group post.

(23-Aug-2020) Version 0.188 of the metamath program adds keywords Conclusion, Fact, Introduction, Paragraph, Scolia, Scolion, Subsection, and Table to bibliographic references. See 'help write bibliography' for the complete current list.

   Older news...

Color key:   Metamath Proof Explorer  Metamath Proof Explorer   Hilbert Space Explorer  Hilbert Space Explorer   User Mathboxes  User Mathboxes  

Last updated on 18-Feb-2026 at 5:35 AM ET.
Recent Additions to the Metamath Proof Explorer   Notes (last updated 7-Dec-2020 )
DateLabelDescription
Theorem
 
11-Feb-2026nfexa2 42331 An inner universal quantifier's variable is bound. (Contributed by SN, 11-Feb-2026.)
Ⅎ𝑥∃𝑦∀𝑥𝜑
 
11-Feb-2026nfale2 42330 An inner existential quantifier's variable is bound. (Contributed by SN, 11-Feb-2026.)
Ⅎ𝑥∀𝑦∃𝑥𝜑
 
11-Feb-2026nfe2 42329 An inner existential quantifier's variable is bound. (Contributed by SN, 11-Feb-2026.)
Ⅎ𝑥∃𝑦∃𝑥𝜑
 
11-Feb-2026nfalh 42328 Version of nfal 2326 with an 'h' hypothesis, avoiding ax-12 2182. (Contributed by SN, 11-Feb-2026.)
(𝜑 → ∀𝑥𝜑)    ⇒   Ⅎ𝑥∀𝑦𝜑
 
11-Feb-2026nfexhe 42327 Version of nfex 2327 with the existential dual to the 'h' hypothesis, avoiding ax-12 2182. (Contributed by SN, 11-Feb-2026.)
(∃𝑥𝜑 → 𝜑)    ⇒   Ⅎ𝑥∃𝑦𝜑
 
10-Feb-2026enssdom 8905 Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.) (Proof shortened by TM, 10-Feb-2026.)
≈ ⊆ ≼
 
10-Feb-2026f1oi 6806 A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) Avoid ax-12 2182. (Revised by TM, 10-Feb-2026.)
( I ↾ 𝐴):𝐴–1-1-onto→𝐴
 
5-Feb-2026dfsb 2069 Simplify definition df-sb 2068 by removing its provable hypothesis. (Contributed by Wolf Lammen, 5-Feb-2026.)
([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
 
4-Feb-2026ss2rabd 4021 Subclass of a restricted class abstraction (deduction form). Saves ax-10 2146, ax-11 2162, ax-12 2182 over using ss2rab 4018 and sylibr 234. (Contributed by SN, 4-Feb-2026.)
(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})
 
4-Feb-2026sbt 2071 A substitution into a theorem yields a theorem. See sbtALT 2074 for a shorter proof requiring more axioms. See chvar 2397 and chvarv 2398 for versions using implicit substitution. (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Jul-2018.) Revise df-sb 2068. (Revised by Steven Nguyen, 6-Jul-2023.) Revise df-sb 2068 again. (Revised by Wolf Lammen, 4-Feb-2026.)
𝜑    ⇒   [𝑡 / 𝑥]𝜑
 
4-Feb-2026sbtlem 2070 In the case of sbt 2071, the hypothesis in df-sb 2068 is derivable from propositional axioms and ax-gen 1796 alone. The essential proof step is presented in this lemma. (Contributed by Wolf Lammen, 4-Feb-2026.)
𝜑    ⇒   ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
 
4-Feb-2026df-sb 2068 Define proper substitution. For our notation, we use [𝑡 / 𝑥]𝜑 to mean "the wff that results from the proper substitution of 𝑡 for 𝑥 in the wff 𝜑". That is, 𝑡 properly replaces 𝑥. For example, [𝑡 / 𝑥]𝑧 ∈ 𝑥 is the same as 𝑧 ∈ 𝑡 (when 𝑥 and 𝑧 are distinct), as shown in elsb2 2130.

Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "𝜑(𝑡) is the wff that results when 𝑡 is properly substituted for 𝑥 in 𝜑(𝑥)". For example, if the original 𝜑(𝑥) is 𝑥 = 𝑡, then 𝜑(𝑡) is 𝑡 = 𝑡, from which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does 𝜑(𝑥) mean? Curry's notation solves this problem.

A very similar notation, namely (𝑦 ∣ 𝑥)𝜑, was introduced in Bourbaki's Set Theory (Chapter 1, Description of Formal Mathematic, 1953).

In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see Theorems sbequ 2088, sbcom2 2178 and sbid2v 2511).

Note that our definition is valid even when 𝑥 and 𝑡 are replaced with the same variable, as sbid 2260 shows. We achieve this by applying twice Tarski's definition sb6 2090 which is valid for disjoint variables, and introducing a dummy variable 𝑦 which isolates 𝑥 from 𝑡, as in dfsb7 2283 with respect to sb5 2280. We can also achieve this by having 𝑥 free in the first conjunct and bound in the second, as the alternate definition dfsb1 2483 shows. Another version that mixes free and bound variables is dfsb3 2496. When 𝑥 and 𝑡 are distinct, we can express proper substitution with the simpler expressions of sb5 2280 and sb6 2090.

Note that the occurrences of a given variable in the definiens are either all bound (𝑥, 𝑦) or all free (𝑡). Also note that the definiens uses only primitive symbols.

This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a disjoint variable condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row.

The hypothesis asserts that the definition is independent of the particular choice of the dummy variable 𝑦. Without this hypothesis, sbjust 2066 would be derivable from propositional axioms alone: one could apply the definiens for [𝑡 / 𝑥]𝜑 twice, using different dummy variables 𝑦 and 𝑧, and then invoke bitr3i 277 to establish their equivalence. This would jeopardize the independence of axioms, as demonstrated in an analoguous situation involving df-ss 3915 to prove ax-8 2115 (see in-ax8 36289).

Prefer dfsb 2069 unless you can prove the hypothesis from fewer axioms in special cases, see sbt 2071. (Contributed by NM, 10-May-1993.) Revised from the original definition dfsb1 2483. (Revised by BJ, 22-Dec-2020.) Add the justification hypothesis. (Revised by Wolf Lammen, 4-Feb-2026.)

(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))    ⇒   ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
 
2-Feb-2026mptelee 28874 A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by SN, 2-Feb-2026.)
(𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ))
 
2-Feb-2026moabex 5401 "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.) Avoid axioms. (Revised by SN, 2-Feb-2026.)
(∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)
 
2-Feb-2026iunss 4995 Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Avoid ax-10 2146, ax-12 2182. (Revised by SN, 2-Feb-2026.)
(∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
 
2-Feb-2026iunssf 4993 Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) Avoid ax-10 2146. (Revised by SN, 2-Feb-2026.)
Ⅎ𝑥𝐶    ⇒   (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
 
2-Feb-2026ss2abim 4009 Class abstractions in a subclass relationship. Reverse direction of ss2ab 4010 which requires fewer axioms. (Contributed by SN, 2-Feb-2026.)
(∀𝑥(𝜑 → 𝜓) → {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓})
 
1-Feb-2026xp0 5719 The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.) Avoid axioms. (Revised by TM, 1-Feb-2026.)
(𝐴 × ∅) = ∅
 
1-Feb-2026uni0 4886 The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Contributed by NM, 16-Sep-1993.) Remove use of ax-nul 5246. (Revised by Eric Schmidt, 4-Apr-2007.) Avoid ax-11 2162. (Revised by TM, 1-Feb-2026.)
∪ ∅ = ∅
 
1-Feb-2026rabss2 4026 Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid axioms. (Revised by TM, 1-Feb-2026.)
(𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑})
 
1-Feb-2026ss2rabdv 4024 Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.) Avoid axioms. (Revised by TM, 1-Feb-2026.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})
 
31-Jan-2026cnv0 6091 The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5236, ax-nul 5246, ax-pr 5372. (Revised by KP, 25-Oct-2021.) Avoid ax-12 2182. (Revised by TM, 31-Jan-2026.)
◡∅ = ∅
 
30-Jan-2026chnsuslle 47003 Length of a subsequence is bounded by the length of original chain. (Contributed by Ender Ting, 30-Jan-2026.)
(𝜑 → 𝑊 ∈ ( < Chain 𝐴))    &   (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))    &   (𝜑 → < Po 𝐴)    ⇒   (𝜑 → (♯‘(𝑊 ∘ 𝐼)) ≤ (♯‘𝑊))
 
30-Jan-2026dfsuccl4 38507 Alternate definition that incorporates the most desirable properties of the successor class. (Contributed by Peter Mazsa, 30-Jan-2026.)
Suc = {𝑛 ∣ ∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)}
 
30-Jan-2026dfsuccl3 38506 Alternate definition of the class of all successors. (Contributed by Peter Mazsa, 30-Jan-2026.)
Suc = {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛}
 
29-Jan-2026nthrucw 47008 Some number sets form a chain of proper subsets. This is rephrasing nthruc 16163 as a statement about chains; the hypothesis sets the ordering relation to be "is a proper subset". The theorem talks about singleton 1, natural numbers, natural-or-zero numbers, integers, rational numbers, algebraic reals (the definition includes complex numbers as algebraic so intersection is taken), real numbers and complex numbers, which are proper subsets in order. (Contributed by Ender Ting, 29-Jan-2026.)
< = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊊ 𝑦}    ⇒   ⟨“{1}ℕℕ0ℤℚ(𝔸 ∩ ℝ)ℝℂ”⟩ ∈ ( < Chain V)
 
29-Jan-2026chner 47007 Any two elements are equivalent in a chain constructed on an equivalence relation. (Contributed by Ender Ting, 29-Jan-2026.)
(𝜑 → ∼ Er 𝐴)    &   (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴))    &   (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))    &   (𝜑 → 𝐼 ∈ (0..^(♯‘𝐶)))    ⇒   (𝜑 → (𝐶‘𝐼) ∼ (𝐶‘𝐽))
 
29-Jan-2026chnerlem3 47006 Lemma for chner 47007- trichotomy of integers within the word's domain. (Contributed by Ender Ting, 29-Jan-2026.)
(𝜑 → ∼ Er 𝐴)    &   (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴))    &   (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))    &   (𝜑 → 𝐼 ∈ (0..^(♯‘𝐶)))    ⇒   (𝜑 → (𝐼 ∈ (0..^𝐽) ∨ 𝐽 ∈ (0..^𝐼) ∨ 𝐼 = 𝐽))
 
29-Jan-2026chnerlem2 47005 Lemma for chner 47007 where the I-th element comes before the J-th. (Contributed by Ender Ting, 29-Jan-2026.)
(𝜑 → ∼ Er 𝐴)    &   (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴))    &   (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))    ⇒   ((𝜑 ∧ 𝐼 ∈ (0..^𝐽)) → (𝐶‘𝐼) ∼ (𝐶‘𝐽))
 
29-Jan-2026chnerlem1 47004 In a chain constructed on an equivalence relation, the last element is equivalent to any. This theorem is a translation of chnub 18530 to equivalence relations. (Contributed by Ender Ting, 29-Jan-2026.)
(𝜑 → ∼ Er 𝐴)    &   (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴))    &   (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))    ⇒   (𝜑 → (𝐶‘𝐽) ∼ (lastS‘𝐶))
 
29-Jan-2026dfblockliftfix2 38756 Alternate definition of the equilibrium / fixed-point condition for "block carriers", cf. df-blockliftfix 38514. (Contributed by Peter Mazsa, 29-Jan-2026.)
BlockLiftFix = ({⟨𝑟, 𝑎⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎} ↾ Rels )
 
29-Jan-2026dfsuccl2 38503 Alternate definition of the class of all successors. (Contributed by Peter Mazsa, 29-Jan-2026.)
Suc = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}
 
29-Jan-2026dfblockliftmap2 38494 Alternate definition of the block lift map. (Contributed by Peter Mazsa, 29-Jan-2026.)
(𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) ↦ ([𝑚]𝑅 × 𝑚))
 
29-Jan-2026dmxrncnvepres2 38477 Domain of the range product with restricted converse epsilon relation. (Contributed by Peter Mazsa, 29-Jan-2026.)
dom (𝑅 ⋉ (◡ E ↾ 𝐴)) = (𝐴 ∩ (dom 𝑅 ∖ {∅}))
 
29-Jan-2026frgr2wwlkeu 30309 For two different vertices in a friendship graph, there is exactly one third vertex being the middle vertex of a (simple) path/walk of length 2 between the two vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 4-Jan-2022.) (Revised by Ender Ting, 29-Jan-2026.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
 
29-Jan-2026usgr2wspthon 29948 A simple path of length 2 between two vertices corresponds to two adjacent edges in a simple graph. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-May-2021.) (Revised by Ender Ting, 29-Jan-2026.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 ((𝑇 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))))
 
29-Jan-2026usgr2wspthons3 29947 A simple path of length 2 between two vertices represented as length 3 string corresponds to two adjacent edges in a simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 16-Mar-2022.) (Revised by Ender Ting, 29-Jan-2026.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
 
29-Jan-2026wpthswwlks2on 29944 For two different vertices, a walk of length 2 between these vertices is a simple path of length 2 between these vertices in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 16-Mar-2022.) (Revised by Ender Ting, 29-Jan-2026.)
((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))
 
29-Jan-2026elwspths2onw 29943 A simple path of length 2 between two vertices (in a simple pseudograph) as length 3 string. This theorem avoids the Axiom of Choice for its proof, at the cost of requiring a simple graph; the more general version is elwspths2on 29942. (Contributed by Ender Ting, 29-Jan-2026.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
 
29-Jan-2026usgrwwlks2on 29938 A walk of length 2 between two vertices as word in a simple graph. This theorem is analogous to umgrwwlks2on 29939 except it talks about simple graphs and therefore does not require the Axiom of Choice for its proof. (Contributed by Ender Ting, 29-Jan-2026.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
 
28-Jan-2026dfsucmap4 38498 Alternate definition of the successor map. (Contributed by Peter Mazsa, 28-Jan-2026.)
SucMap = (𝑚 ∈ V ↦ suc 𝑚)
 
28-Jan-2026dfsucmap2 38497 Alternate definition of the successor map. (Contributed by Peter Mazsa, 28-Jan-2026.)
SucMap = ( I AdjLiftMap dom I )
 
28-Jan-2026dfsucmap3 38496 Alternate definition of the successor map. (Contributed by Peter Mazsa, 28-Jan-2026.)
SucMap = ( I AdjLiftMap V)
 
28-Jan-2026blockadjliftmap 38492 A "two-stage" construction is obtained by first forming the block relation (𝑅 ⋉ ◡ E ) and then adjoining elements as "BlockAdj". Combined, it uses the relation ((𝑅 ⋉ ◡ E ) ∪ ◡ E ). (Contributed by Peter Mazsa, 28-Jan-2026.)
((𝑅 ⋉ ◡ E ) AdjLiftMap 𝐴) = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))}
 
28-Jan-2026dfadjliftmap2 38491 Alternate definition of the adjoined lift map. (Contributed by Peter Mazsa, 28-Jan-2026.)
(𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))
 
28-Jan-2026ecuncnvepres 38439 The restricted union with converse epsilon relation coset of 𝐵. (Contributed by Peter Mazsa, 28-Jan-2026.)
(𝐵 ∈ 𝐴 → [𝐵]((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐵 ∪ [𝐵]𝑅))
 
28-Jan-2026ecunres 38438 The restricted union coset of 𝐵. (Contributed by Peter Mazsa, 28-Jan-2026.)
(𝐵 ∈ 𝑉 → [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))
 
28-Jan-2026ecun 38437 The union coset of 𝐴. (Contributed by Peter Mazsa, 28-Jan-2026.)
(𝐴 ∈ 𝑉 → [𝐴](𝑅 ∪ 𝑆) = ([𝐴]𝑅 ∪ [𝐴]𝑆))
 
28-Jan-2026dmxrnuncnvepres 38436 Domain of the range Cartesian product with the converse epsilon relation combined with the union with the converse epsilon, restricted. (Contributed by Peter Mazsa, 28-Jan-2026.)
dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∖ {∅})
 
28-Jan-2026dmuncnvepres 38435 Domain of the union with the converse epsilon, restricted. (Contributed by Peter Mazsa, 28-Jan-2026.)
dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))
 
28-Jan-2026dmcnvepres 38434 Domain of the restricted converse epsilon relation. (Contributed by Peter Mazsa, 28-Jan-2026.)
dom (◡ E ↾ 𝐴) = (𝐴 ∖ {∅})
 
28-Jan-2026sps3wwlks2on 29937 A length 3 string which represents a walk of length 2 between two vertices. Concerns simple pseudographs, in contrast to s3wwlks2on 29936 and does not require the Axiom of Choice for its proof. (Contributed by Ender Ting, 28-Jan-2026.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
 
27-Jan-2026sucpre 38529 suc is a right-inverse of pre on Suc. This theorem states the partial inverse relation in the direction we most often need. (Contributed by Peter Mazsa, 27-Jan-2026.)
(𝑁 ∈ Suc → suc pre 𝑁 = 𝑁)
 
27-Jan-2026eupre 38526 Unique predecessor exists on the successor class. (Contributed by Peter Mazsa, 27-Jan-2026.)
(𝑁 ∈ 𝑉 → (𝑁 ∈ Suc ↔ ∃!𝑚 𝑚 SucMap 𝑁))
 
27-Jan-2026dfpre 38509 Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 27-Jan-2026.)
pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , V, 𝑁))
 
27-Jan-2026df-pre 38508 Define the term-level successor-predecessor. It is the unique 𝑚 with suc 𝑚 = 𝑁 when such an 𝑚 exists; otherwise pre 𝑁 is the arbitrary default chosen by ℩. See its alternate definitions dfpre 38509, dfpre2 38510, dfpre3 38511 and dfpre4 38513.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6253 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6265, cf. also df-bnj14 34722) in several respects. Its most abstract property as a specialisation is that it has a unique existing value by default. This is in contrast to the general version. The uniqueness (conditional on existence) is implied by the property of this specific instance of the general case involving the successor map df-sucmap 38495 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 38522, which originates from suc11reg 9516. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5835.

Note that dom SucMap = V (see dmsucmap 38501), so the equivalent definition dfpre 38509 uses (℩𝑚𝑚 ∈ Pred( SucMap , V, 𝑁)). (Contributed by Peter Mazsa, 27-Jan-2026.)

pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
 
26-Jan-2026dfpre4 38513 Alternate definition of the predecessor of the 𝑁 set. The ◡ SucMap is just the "PreMap"; we did not define it because we do not expect to use it extensively in future (cf. the comments of df-sucmap 38495). (Contributed by Peter Mazsa, 26-Jan-2026.)
(𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))
 
26-Jan-2026dfpred4 38512 Alternate definition of the predecessor class when 𝑁 is a set. (Contributed by Peter Mazsa, 26-Jan-2026.)
(𝑁 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑁) = [𝑁]◡(𝑅 ↾ 𝐴))
 
25-Jan-2026df-shiftstable 38515 Define shift-stability, a general "procedure" pattern for "the one-step backward shift/transport of 𝐹 along 𝑆", and then ∩ 𝐹 enforces "and it already holds here".

Let 𝐹 be a relation encoding a property that depends on a "level" coordinate (for example, a feasibility condition indexed by a carrier, a grade, or a stage in a construction). Let 𝑆 be a shift relation between levels (for example, the successor map SucMap, or any other grading step).

The composed relation (𝑆 ∘ 𝐹) transports 𝐹 one step along the shift: 𝑟(𝑆 ∘ 𝐹)𝑛 means there exists a predecessor level 𝑚 such that 𝑟𝐹𝑚 and 𝑚𝑆𝑛 (e.g., 𝑚 SucMap 𝑛). We do not introduce a separate notation for "Shift" because it is simply the standard relational composition df-co 5628.

The intersection ((𝑆 ∘ 𝐹) ∩ 𝐹) is the locally shift-stable fragment of 𝐹: it consists exactly of those points where the property holds at some immediate predecessor that shifts to 𝑛 and also holds at level 𝑛. In other words, it isolates the part of 𝐹 that is already compatible with one-step tower coherence.

This definition packages a common construction pattern used throughout the development: "constrain by one-step stability under a chosen shift, then additionally constrain by 𝐹". Iterating the operator (𝑋 ↦ ((𝑆 ∘ 𝑋) ∩ 𝑋) corresponds to multi-step/tower coherence; the one-step definition here is the economical kernel from which such "tower" readings can be developed when needed. (Contributed by Peter Mazsa, 25-Jan-2026.)

(𝑆 ShiftStable 𝐹) = ((𝑆 ∘ 𝐹) ∩ 𝐹)
 
25-Jan-2026df-blockliftfix 38514 Define the equilibrium / fixed-point condition for "block carriers".

Start with a candidate block-family 𝑎 (a set whose elements you intend to treat as blocks). Combine it with a relation 𝑟 by forming the block-lift span 𝑇 = (𝑟 ⋉ (◡ E ↾ 𝑎)). For a block 𝑢 ∈ 𝑎, the fiber [𝑢]𝑇 is the set of all outputs produced from "external targets" of 𝑟 together with "internal members" of 𝑢; in other words, 𝑇 is the mechanism that generates new blocks from old ones.

Now apply the standard quotient construction (dom 𝑇 / 𝑇). This produces the family of all T-blocks (the cosets [𝑥]𝑇 of witnesses 𝑥 in the domain of 𝑇). In general, this operation can change your carrier: starting from 𝑎, it may generate a different block-family (dom 𝑇 / 𝑇).

The equation (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎 says exactly: if you generate blocks from 𝑎 using the lift determined by 𝑟 (cf. df-blockliftmap 38493), you get back the same 𝑎. So 𝑎 is stable under the block-generation operator induced by 𝑟. This is why it is a genuine fixpoint/equilibrium condition: one application of the "make-the-blocks" operator causes no carrier drift, i.e. no hidden refinement/coarsening of what counts as a block.

Here, we generate this from the df-blockliftmap 38493, taking the range of the two sides, resulting in (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) (via dfqs2 42355), which you can define as "( R BlockLift A )" . In that case, you can define BlockLiftFix as "{ <. r , a >. | ( r BlockLift a ) = a }", or typed as "{ <. r , a >. | ( r e. Rels /\ ( r BlockLift a ) = a ) }".

This is a relation-typed equilibrium predicate. Restricting it to 𝑟 ∈ Rels (see the explicit restriction in the alternate definition dfblockliftfix2 38756) prevents representation junk (which may contain non-ordered-pair 𝑟 that would not affect the predicate 𝑥𝑟𝑦, because that predicate only looks at ordered pairs) and makes the module composable with later Rels-based infrastructure; sethood of the quotient does not require it in itself. (Contributed by Peter Mazsa, 25-Jan-2026.)

BlockLiftFix = {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎)}
 
25-Jan-2026df-succl 38502 Define Suc as the class of all successors, i.e. the range of the successor map: 𝑛 ∈ Suc iff ∃𝑚suc 𝑚 = 𝑛 (see dfsuccl2 38503). By injectivity of suc (suc11reg 9516), every 𝑛 ∈ Suc has at most one predecessor, which is exactly what pre 𝑛 (df-pre 38508) names. Cf. dfsuccl3 38506 and dfsuccl4 38507. (Contributed by Peter Mazsa, 25-Jan-2026.)
Suc = ran SucMap
 
25-Jan-2026df-sucmap 38495 Define the successor map, directly as the graph of the successor operation, using only elementary set theory (ordered-pair class abstraction). This avoids committing to any particular construction of the successor function/class from other operators (e.g. a union/composition presentation), while remaining provably equivalent to those presentations (cf. dfsucmap2 38497 and dfsucmap3 38496 vs. df-succf 35935 and dfsuccf2 36006). For maximum mappy shape, see dfsucmap4 38498.

We also treat the successor relation as the default shift relation for grading/tower arguments (cf. df-shiftstable 38515). Because it is used pervasively in shift-lift infrastructure, we adopt the short name SucMap rather than the fully systematic "SucAdjLiftMap".

You may also define the predecessor relation as the converse graph "PreMap" as ◡ SucMap, which reverses successor edges ( cf. cnvopab 6088) and sends each successor to its (unique) predecessor when it exists. (Contributed by Peter Mazsa, 25-Jan-2026.)

SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
 
25-Jan-2026ecxrncnvep2 38454 The (𝑅 ⋉ ◡ E )-coset of a set is the Cartesian product of its 𝑅-coset and the set. (Contributed by Peter Mazsa, 25-Jan-2026.)
(𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ ◡ E ) = ([𝐴]𝑅 × 𝐴))
 
25-Jan-2026r1omfv 35142 Value of the cumulative hierarchy of sets function at ω. (Contributed by BTernaryTau, 25-Jan-2026.)
(𝑅1‘ω) = ∪ (𝑅1 “ ω)
 
25-Jan-2026r12 35127 Value of the cumulative hierarchy of sets function at 2o. (Contributed by BTernaryTau, 25-Jan-2026.)
(𝑅1‘2o) = 2o
 
25-Jan-2026esplyind 33613 A recursive formula for the elementary symmetric polynomials. (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝑊 = (𝐼 mPoly 𝑅)    &   𝑉 = (𝐼 mVar 𝑅)    &    + = (+g‘𝑊)    &    · = (.r‘𝑊)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   𝐺 = ((𝐼extendVars𝑅)‘𝑌)    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑌 ∈ 𝐼)    &   𝐽 = (𝐼 ∖ {𝑌})    &   𝐸 = (𝐽eSymPoly𝑅)    &   (𝜑 → 𝐾 ∈ (1...(♯‘𝐼)))    &   𝐶 = {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}    ⇒   (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (((𝑉‘𝑌) · (𝐺‘(𝐸‘(𝐾 − 1)))) + (𝐺‘(𝐸‘𝐾))))
 
25-Jan-2026esplyfval3 33612 Alternate expression for the value of the 𝐾-th elementary symmetric polynomial. (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐾 ∈ ℕ0)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    ⇒   (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))
 
25-Jan-2026esplyfval2 33605 When 𝐾 is out-of-bounds, the 𝐾-th elementary symmetric polynomial is zero. (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐾 ∈ (ℕ0 ∖ (0...(♯‘𝐼))))    &   𝑍 = (0g‘(𝐼 mPoly 𝑅))    ⇒   (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = 𝑍)
 
25-Jan-2026mplmulmvr 33590 Multiply a polynomial 𝐹 with a variable 𝑋 (i.e. with a monic monomial). (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑋 = ((𝐼 mVar 𝑅)‘𝑌)    &   𝑀 = (Base‘𝑃)    &    · = (.r‘𝑃)    &    0 = (0g‘𝑅)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   𝐴 = ((𝟭‘𝐼)‘{𝑌})    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑌 ∈ 𝐼)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐹 ∈ 𝑀)    ⇒   (𝜑 → (𝑋 · 𝐹) = (𝑏 ∈ 𝐷 ↦ if((𝑏‘𝑌) = 0, 0 , (𝐹‘(𝑏 ∘f − 𝐴)))))
 
25-Jan-2026mvrvalind 33589 Value of the generating elements of the power series structure, expressed using the indicator function. (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝑉 = (𝐼 mVar 𝑅)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑊)    &   (𝜑 → 𝑅 ∈ 𝑌)    &   (𝜑 → 𝑋 ∈ 𝐼)    &   (𝜑 → 𝐹 ∈ 𝐷)    &   𝐴 = ((𝟭‘𝐼)‘{𝑋})    ⇒   (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = 𝐴, 1 , 0 ))
 
25-Jan-2026extvfvalf 33588 The "variable extension" function maps polynomials with variables indexed in 𝐽 to polynomials with variables indexed in 𝐼. (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &    0 = (0g‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &   𝐽 = (𝐼 ∖ {𝐴})    &   𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   (𝜑 → 𝐴 ∈ 𝐼)    &   𝑁 = (Base‘(𝐼 mPoly 𝑅))    ⇒   (𝜑 → ((𝐼extendVars𝑅)‘𝐴):𝑀⟶𝑁)
 
25-Jan-2026extvfvcl 33587 Closure for the "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &    0 = (0g‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &   𝐽 = (𝐼 ∖ {𝐴})    &   𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   (𝜑 → 𝐴 ∈ 𝐼)    &   (𝜑 → 𝐹 ∈ 𝑀)    &   𝑁 = (Base‘(𝐼 mPoly 𝑅))    ⇒   (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ 𝑁)
 
25-Jan-2026extvfvvcl 33586 Closure for the "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &    0 = (0g‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &   𝐽 = (𝐼 ∖ {𝐴})    &   𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   (𝜑 → 𝐴 ∈ 𝐼)    &   (𝜑 → 𝐹 ∈ 𝑀)    &   (𝜑 → 𝑋 ∈ 𝐷)    ⇒   (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) ∈ 𝐵)
 
25-Jan-2026extvfvv 33585 The "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &    0 = (0g‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   (𝜑 → 𝐴 ∈ 𝐼)    &   𝐽 = (𝐼 ∖ {𝐴})    &   𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   (𝜑 → 𝐹 ∈ 𝑀)    &   (𝜑 → 𝑋 ∈ 𝐷)    ⇒   (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) = if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ))
 
25-Jan-2026extvfv 33584 The "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &    0 = (0g‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   (𝜑 → 𝐴 ∈ 𝐼)    &   𝐽 = (𝐼 ∖ {𝐴})    &   𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   (𝜑 → 𝐹 ∈ 𝑀)    ⇒   (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
 
25-Jan-2026extvfval 33583 The "variable extension" function evaluated for adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &    0 = (0g‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   (𝜑 → 𝐴 ∈ 𝐼)    &   𝐽 = (𝐼 ∖ {𝐴})    &   𝑀 = (Base‘(𝐽 mPoly 𝑅))    ⇒   (𝜑 → ((𝐼extendVars𝑅)‘𝐴) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))))
 
25-Jan-2026extvval 33582 Value of the "variable extension" function. (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &    0 = (0g‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐽 = (𝐼 ∖ {𝑎})    &   𝑀 = (Base‘(𝐽 mPoly 𝑅))    ⇒   (𝜑 → (𝐼extendVars𝑅) = (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))))
 
25-Jan-2026indconst1 32847 Indicator of the whole set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
(𝑂 ∈ 𝑉 → ((𝟭‘𝑂)‘𝑂) = (𝑂 × {1}))
 
25-Jan-2026indconst0 32846 Indicator of the empty set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
(𝑂 ∈ 𝑉 → ((𝟭‘𝑂)‘∅) = (𝑂 × {0}))
 
25-Jan-2026nn0diffz0 32781 Upper set of the nonnegative integers. (Contributed by Thierry Arnoux, 25-Jan-2026.)
(𝑁 ∈ ℕ0 → (ℕ0 ∖ (0...𝑁)) = (ℤ≥‘(𝑁 + 1)))
 
25-Jan-2026rnressnsn 32662 The range of a restriction to a singleton is a singleton. See dmressnsn 5976. (Contributed by Thierry Arnoux, 25-Jan-2026.)
((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ran (𝐹 ↾ {𝐴}) = {(𝐹‘𝐴)})
 
25-Jan-2026partfun2 32661 Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. See also partfun 6633 and ifmpt2v 7454. (Contributed by Thierry Arnoux, 25-Jan-2026.)
𝐷 = {𝑥 ∈ 𝐴 ∣ 𝜑}    ⇒   (𝑥 ∈ 𝐴 ↦ if(𝜑, 𝐵, 𝐶)) = ((𝑥 ∈ 𝐷 ↦ 𝐵) ∪ (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶))
 
25-Jan-2026tz6.12-2 6815 Function value when 𝐹 is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2146, ax-11 2162, ax-12 2182. (Revised by TM, 25-Jan-2026.)
(¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
 
24-Jan-2026df-blockliftmap 38493 Define the block lift map. Given a relation 𝑅 and a carrier/set 𝐴, we form the block relation (𝑅 ⋉ ◡ E ) (i.e., "follow both 𝑅 and element"), restricted to 𝐴 (or, equivalently, "follow both 𝑅 and elements-of-A", cf. xrnres2 38470). Then map each domain element 𝑚 to its coset [𝑚] under that restricted block relation.

For 𝑚 in the domain, which requires (𝑚 ∈ 𝐴 ∧ 𝑚 ≠ ∅ ∧ [𝑚]𝑅 ≠ ∅) (cf. eldmxrncnvepres 38478), the fiber has the product form [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚), so the block relation lifts a block 𝑚 to the rectangular grid "external labels × internal members", see dfblockliftmap2 38494. Contrast: while the adjoined lift, via (𝑅 ∪ ◡ E ), attaches neighbors and members in a single relation (see dfadjliftmap2 38491), the block lift labels each internal member by each external neighbor.

For the general case and a two-stage construction (first block lift, then adjoin membership), see the comments to df-adjliftmap 38490. For the equilibrium condition, see df-blockliftfix 38514 and dfblockliftfix2 38756. (Contributed by Peter Mazsa, 24-Jan-2026.)

(𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴)))
 
24-Jan-2026df-adjliftmap 38490 Define the adjoined lift map. Given a relation 𝑅 and a carrier/set 𝐴, we form the adjoined relation (𝑅 ∪ ◡ E ) (i.e., "follow 𝑅 or follow elements"), restricted to 𝐴, and map each domain element 𝑚 to its coset [𝑚] under that restricted adjoined relation. Thus, for 𝑚 in its domain, we have (𝑚 ∪ [𝑚]𝑅), see dfadjliftmap2 38491.

Its key special case is successor: for 𝑅 = I and 𝐴 = dom I, or 𝐴 = V, the adjoined relation is ( I ∪ ◡ E ), and the coset becomes [𝑚]( I ∪ ◡ E ) = (𝑚 ∪ {𝑚}). So ( I AdjLiftMap dom I ) or ( I AdjLiftMap V) (see dfsucmap2 38497 and dfsucmap3 38496) are exactly the successor map 𝑚 ↦ suc 𝑚 (cf. dfsucmap4 38498), which is a prerequisite for accepting the adjoining lift as the right generalization of successor.

A maximally generic form would be "( R F LiftMap A )" defined as (𝑚 ∈ dom ((𝑅𝐹◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅𝐹◡ E ) ↾ 𝐴)) where 𝐹 is an object-level binary operator on relations (used via df-ov 7355). However, ∪ and ⋉ are introduced in set.mm as class constructors (e.g. df-un 3903), not as an object-level binary function symbol 𝐹 that can be passed as a parameter. To make the generic 𝐹-pattern literally usable, we would need to reify union and ⋉ as function-objects, which is additional infrastructure. To avoid introducing operator-as-function objects solely to support 𝐹, we define:

AdjLiftMap directly using df-un 3903, and

BlockLiftMap directly using the existing ⋉ constructor dfxrn2 38429,

so we treat any "generic 𝐹-LiftMap" as optional future generalization, not a dependency.

We prefer to avoid defining too many concepts. For this reason, we will not introduce

a named "adjoining relation",

a named carrier "adjoining lift" "( R AdjLift A )", in place of ran (𝑅 AdjLiftMap 𝐴), which is (dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) / ((𝑅 ∪ ◡ E ) ↾ 𝐴)), cf. dfqs2 42355,

or the equilibrium condition "AdjLiftFix" , in place of {⟨𝑟, 𝑎⟩ ∣ (dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) / ((𝑅 ∪ ◡ E ) ↾ 𝐴)) = 𝑎} (cf. its analog df-blockliftfix 38514 and dfblockliftfix2 38756). These are definable by simple expansions and/or domain-quotient theorems when needed.

A "two-stage" construction is obtained by first forming the block relation (𝑅 ⋉ ◡ E ) and then adjoining elements as "BlockAdj" . Combined, it uses the relation ((𝑅 ⋉ ◡ E ) ∪ ◡ E ), which for 𝑚 in its domain (𝐴 ∖ {∅}) gives (𝑚 ∪ [𝑚](𝑅 ⋉ ◡ E )), yielding "BlockAdjLiftMap" (cf. blockadjliftmap 38492) and "BlockAdjLiftFix". We only introduce these if a downstream theorem actually requires them. (Contributed by Peter Mazsa, 24-Jan-2026.)

(𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))
 
24-Jan-2026r1omhfb 35144 The class of all hereditarily finite sets is the only class with the property that all sets are members of it iff they are finite and all of their elements are members of it. (Contributed by BTernaryTau, 24-Jan-2026.)
(𝐻 = ∪ (𝑅1 “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
 
24-Jan-2026trssfir1om 35143 If every element in a transitive class is finite, then every element is also hereditarily finite. (Contributed by BTernaryTau, 24-Jan-2026.)
((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ ∪ (𝑅1 “ ω))
 
24-Jan-2026r11 35126 Value of the cumulative hierarchy of sets function at 1o. (Contributed by BTernaryTau, 24-Jan-2026.)
(𝑅1‘1o) = 1o
 
24-Jan-2026rnco 6204 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) Avoid ax-11 2162. (Revised by TM, 24-Jan-2026.)
ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
 
24-Jan-2026dm0rn0 5868 An empty domain is equivalent to an empty range. (Contributed by NM, 21-May-1998.) Avoid ax-10 2146, ax-11 2162, ax-12 2182. (Revised by TM, 24-Jan-2026.)
(dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
 
24-Jan-2026eqabcbw 2807 Version of eqabcb 2873 using implicit substitution, which requires fewer axioms. (Contributed by TM, 24-Jan-2026.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑦(𝜓 ↔ 𝑦 ∈ 𝐴))
 
24-Jan-2026excomw 2047 Weak version of excom 2167 and biconditional form of excomimw 2045. Uses only Tarski's FOL axiom schemes. (Contributed by TM, 24-Jan-2026.)
(𝑥 = 𝑤 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))    ⇒   (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
 
22-Jan-2026chnsubseq 47002 An order-preserving subsequence of an ordered chain is itself a chain. (Contributed by Ender Ting, 22-Jan-2026.)
(𝜑 → 𝑊 ∈ ( < Chain 𝐴))    &   (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))    &   (𝜑 → < Po 𝐴)    ⇒   (𝜑 → (𝑊 ∘ 𝐼) ∈ ( < Chain 𝐴))
 
22-Jan-2026chnsubseqwl 47001 A subsequence of a chain has the same length as its indexing sequence. (Contributed by Ender Ting, 22-Jan-2026.)
(𝜑 → 𝑊 ∈ ( < Chain 𝐴))    &   (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))    ⇒   (𝜑 → (♯‘(𝑊 ∘ 𝐼)) = (♯‘𝐼))
 
22-Jan-2026chnsubseqword 47000 A subsequence of a chain is a word. (Contributed by Ender Ting, 22-Jan-2026.)
(𝜑 → 𝑊 ∈ ( < Chain 𝐴))    &   (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))    ⇒   (𝜑 → (𝑊 ∘ 𝐼) ∈ Word 𝐴)
 
22-Jan-2026r1filim 35136 A finite set appears in the cumulative hierarchy prior to a limit ordinal iff all of its elements appear in the cumulative hierarchy prior to that limit ordinal. (Contributed by BTernaryTau, 22-Jan-2026.)
((𝐴 ∈ Fin ∧ Lim 𝐵) → (𝐴 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵)))
 
22-Jan-2026rankfilimb 35134 The rank of a finite well-founded set is less than a limit ordinal iff the ranks of all of its elements are less than that limit ordinal. (Contributed by BTernaryTau, 22-Jan-2026.)
((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))
 
21-Jan-2026r1omhfbregs 35154 The class of all hereditarily finite sets is the only class with the property that all sets are members of it iff they are finite and all of their elements are members of it. This version of r1omhfb 35144 replaces setinds2 9648 with setinds2regs 35150 and trssfir1om 35143 with trssfir1omregs 35153. (Contributed by BTernaryTau, 21-Jan-2026.)
(𝐻 = ∪ (𝑅1 “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
 
20-Jan-2026trssfir1omregs 35153 If every element in a transitive class is finite, then every element is also hereditarily finite. This version of trssfir1om 35143 replaces setinds2 9648 with setinds2regs 35150. (Contributed by BTernaryTau, 20-Jan-2026.)
((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ ∪ (𝑅1 “ ω))
 
20-Jan-2026df-extv 33581 Define the "variable extension" function. The function ((𝐼extendVars𝑅)‘𝐴) converts polynomials with variables indexed by (𝐼 ∖ {𝐴}) into polynomials indexed by 𝐼, and therefore maps elements of ((𝐼 ∖ {𝐴}) mPoly 𝑅) onto (𝐼 mPoly 𝑅). (Contributed by Thierry Arnoux, 20-Jan-2026.)
extendVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑎 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0g‘𝑟))))))
 
20-Jan-2026chnfibg 18544 Given a partial order, the set of chains is finite iff the alphabet is finite. (Contributed by Ender Ting, 20-Jan-2026.)
( < Po 𝐴 → (𝐴 ∈ Fin ↔ ( < Chain 𝐴) ∈ Fin))
 
20-Jan-2026chninf 18543 There is an infinite number of chains for any infinite alphabet and any relation. For instance, all the singletons of alphabet characters match. (Contributed by Ender Ting, 20-Jan-2026.)
(𝐴 ∉ Fin → ( < Chain 𝐴) ∉ Fin)
 
20-Jan-2026chnfi 18542 There is a finite number of chains over finite domain, as long as the relation orders it. (Contributed by Ender Ting, 20-Jan-2026.)
((𝐴 ∈ Fin ∧ < Po 𝐴) → ( < Chain 𝐴) ∈ Fin)
 
20-Jan-2026chnpolfz 18541 Provided that chain's relation is a partial order, the chain length is restricted to a specific integer range. (Contributed by Ender Ting, 20-Jan-2026.)
(𝜑 → < Po 𝐴)    &   (𝜑 → 𝐵 ∈ ( < Chain 𝐴))    &   (𝜑 → 𝐴 ∈ Fin)    ⇒   (𝜑 → (♯‘𝐵) ∈ (0...(♯‘𝐴)))
 
20-Jan-2026chnpolleha 18540 A chain under relation which orders the alphabet has at most alphabet's size elements in it. (Contributed by Ender Ting, 20-Jan-2026.)
(𝜑 → < Po 𝐴)    &   (𝜑 → 𝐵 ∈ ( < Chain 𝐴))    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (♯‘𝐵) ≤ (♯‘𝐴))
 
20-Jan-2026chnpoadomd 18539 A chain under relation which orders the alphabet cannot have more elements than the alphabet itself. (Contributed by Ender Ting, 20-Jan-2026.)
(𝜑 → < Po 𝐴)    &   (𝜑 → 𝐵 ∈ ( < Chain 𝐴))    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (0..^(♯‘𝐵)) ≼ 𝐴)
 
20-Jan-2026chnpof1 18538 A chain under relation which orders the alphabet is a one-to-one function from its domain to alphabet. (Contributed by Ender Ting, 20-Jan-2026.)
(𝜑 → < Po 𝐴)    &   (𝜑 → 𝐵 ∈ ( < Chain 𝐴))    ⇒   (𝜑 → 𝐵:(0..^(♯‘𝐵))–1-1→𝐴)
 
20-Jan-2026chnf 18537 A chain is a zero-based finite sequence with a recoverable upper limit. (Contributed by Ender Ting, 20-Jan-2026.)
(𝐵 ∈ ( < Chain 𝐴) → 𝐵:(0..^(♯‘𝐵))⟶𝐴)
 
20-Jan-2026chnrev 18535 Reverse of a chain is chain under the converse relation and same domain. (Contributed by Ender Ting, 20-Jan-2026.)
(𝐵 ∈ ( < Chain 𝐴) → (reverse‘𝐵) ∈ (◡ < Chain 𝐴))
 
20-Jan-2026chnccat 18534 Concatenate two chains. (Contributed by Ender Ting, 20-Jan-2026.)
(𝜑 → 𝑇 ∈ ( < Chain 𝐴))    &   (𝜑 → 𝑈 ∈ ( < Chain 𝐴))    &   (𝜑 → (𝑇 = ∅ ∨ 𝑈 = ∅ ∨ (lastS‘𝑇) < (𝑈‘0)))    ⇒   (𝜑 → (𝑇 ++ 𝑈) ∈ ( < Chain 𝐴))
 
20-Jan-2026chnrdss 18525 Subset theorem for chains. (Contributed by Ender Ting, 20-Jan-2026.)
(( < ⊆ 𝑅 ∧ 𝐴 ⊆ 𝐵) → ( < Chain 𝐴) ⊆ (𝑅 Chain 𝐵))
 
20-Jan-2026chndss 18524 Chains with an alphabet are also chains with any superset alphabet. (Contributed by Ender Ting, 20-Jan-2026.)
(𝐴 ⊆ 𝐵 → ( < Chain 𝐴) ⊆ ( < Chain 𝐵))
 
20-Jan-2026chnrss 18523 Chains under a relation are also chains under any superset relation. (Contributed by Ender Ting, 20-Jan-2026.)
( < ⊆ 𝑅 → ( < Chain 𝐴) ⊆ (𝑅 Chain 𝐴))
 
20-Jan-2026nfchnd 18519 Bound-variable hypothesis builder for chain collection constructor. (Contributed by Ender Ting, 20-Jan-2026.)
(𝜑 → Ⅎ𝑥 < )    &   (𝜑 → Ⅎ𝑥𝐴)    ⇒   (𝜑 → Ⅎ𝑥( < Chain 𝐴))
 
19-Jan-2026r1omhf 35138 A set is hereditarily finite iff it is finite and all of its elements are hereditarily finite. (Contributed by BTernaryTau, 19-Jan-2026.)
(𝐴 ∈ ∪ (𝑅1 “ ω) ↔ (𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ ω)))
 
19-Jan-2026r1filimi 35135 If all elements in a finite set appear in the cumulative hierarchy prior to a limit ordinal, then that set also appears in the cumulative hierarchy prior to the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)
((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))
 
19-Jan-2026rankfilimbi 35133 If all elements in a finite well-founded set have a rank less than a limit ordinal, then the rank of that set is also less than the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)
(((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)
 
19-Jan-2026rankval4b 35132 The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. This variant of rankval4 9767 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅1. (Contributed by BTernaryTau, 19-Jan-2026.)
(𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
 
19-Jan-2026rankval2b 35131 Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. This variant of rankval2 9718 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅1. (Contributed by BTernaryTau, 19-Jan-2026.)
(𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (𝑅1‘𝑥)})
 
19-Jan-2026r1wf 35128 Each stage in the cumulative hierarchy is well-founded. (Contributed by BTernaryTau, 19-Jan-2026.)
(𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On)
 
18-Jan-2026esplysply 33611 The 𝐾-th elementary symmetric polynomial is symmetric. (Contributed by Thierry Arnoux, 18-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))    ⇒   (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ (𝐼SymPoly𝑅))
 
18-Jan-2026esplyfv 33610 Coefficient for the 𝐾-th elementary symmetric polynomial and a bag of variables 𝐹: the coefficient is 1 for the bags of exactly 𝐾 variables, having exponent at most 1. (Contributed by Thierry Arnoux, 18-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))    &   (𝜑 → 𝐹 ∈ 𝐷)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    ⇒   (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((ran 𝐹 ⊆ {0, 1} ∧ (♯‘(𝐹 supp 0)) = 𝐾), 1 , 0 ))
 
18-Jan-2026esplyfv1 33609 Coefficient for the 𝐾-th elementary symmetric polynomial and a bag of variables 𝐹 where variables are not raised to a power. (Contributed by Thierry Arnoux, 18-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))    &   (𝜑 → 𝐹 ∈ 𝐷)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    &   (𝜑 → ran 𝐹 ⊆ {0, 1})    ⇒   (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ))
 
18-Jan-2026esplymhp 33608 The 𝐾-th elementary symmetric polynomial is homogeneous of degree 𝐾. (Contributed by Thierry Arnoux, 18-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐻 = (𝐼 mHomP 𝑅)    ⇒   (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ (𝐻‘𝐾))
 
18-Jan-2026esplympl 33607 Elementary symmetric polynomials are polynomials. (Contributed by Thierry Arnoux, 18-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝑀 = (Base‘(𝐼 mPoly 𝑅))    ⇒   (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ 𝑀)
 
18-Jan-2026esplylem 33606 Lemma for esplyfv 33610 and others. (Contributed by Thierry Arnoux, 18-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐾 ∈ ℕ0)    ⇒   (𝜑 → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
 
18-Jan-2026esplyfval 33604 The 𝐾-th elementary polynomial for a given index 𝐼 of variables and base ring 𝑅. (Contributed by Thierry Arnoux, 18-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   (𝜑 → 𝐾 ∈ ℕ0)    ⇒   (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))
 
18-Jan-2026esplyval 33603 The elementary polynomials for a given index 𝐼 of variables and base ring 𝑅. (Contributed by Thierry Arnoux, 18-Jan-2026.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    ⇒   (𝜑 → (𝐼eSymPoly𝑅) = (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})))))
 
18-Jan-2026issply 33602 Conditions for being a symmetric polynomial. (Contributed by Thierry Arnoux, 18-Jan-2026.)
𝑆 = (SymGrp‘𝐼)    &   𝑃 = (Base‘𝑆)    &   𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   (𝜑 → 𝐹 ∈ 𝑀)    &   (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑥 ∈ 𝐷) → (𝐹‘(𝑥 ∘ 𝑝)) = (𝐹‘𝑥))    ⇒   (𝜑 → 𝐹 ∈ (𝐼SymPoly𝑅))
 
18-Jan-2026df-esply 33599 Define elementary symmetric polynomials. (Contributed by Thierry Arnoux, 18-Jan-2026.)
eSymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘})))))
 
18-Jan-2026gsumind 33317 The group sum of an indicator function of the set 𝐴 gives the size of 𝐴. (Contributed by Thierry Arnoux, 18-Jan-2026.)
(𝜑 → 𝑂 ∈ 𝑉)    &   (𝜑 → 𝐴 ⊆ 𝑂)    &   (𝜑 → 𝐴 ∈ Fin)    ⇒   (𝜑 → (ℂfld Σg ((𝟭‘𝑂)‘𝐴)) = (♯‘𝐴))
 
18-Jan-2026indfsid 32857 Conditions for a function to be an indicator function. (Contributed by Thierry Arnoux, 18-Jan-2026.)
(𝜑 → 𝑂 ∈ 𝑉)    &   (𝜑 → 𝐹:𝑂⟶{0, 1})    ⇒   (𝜑 → 𝐹 = ((𝟭‘𝑂)‘(𝐹 supp 0)))
 
18-Jan-2026indfsd 32856 The indicator function of a finite set has finite support. (Contributed by Thierry Arnoux, 18-Jan-2026.)
(𝜑 → 𝑂 ∈ 𝑉)    &   (𝜑 → 𝐴 ⊆ 𝑂)    &   (𝜑 → 𝐴 ∈ Fin)    ⇒   (𝜑 → ((𝟭‘𝑂)‘𝐴) finSupp 0)
 
18-Jan-2026hashimaf1 32798 Taking the image of a set by a one-to-one function does not affect size. (Contributed by Thierry Arnoux, 18-Jan-2026.)
(𝜑 → 𝐹:𝐴–1-1→𝐵)    &   (𝜑 → 𝐶 ⊆ 𝐴)    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (♯‘(𝐹 “ 𝐶)) = (♯‘𝐶))
 
18-Jan-2026pw2cut2 28383 Cut expression for powers of two. Theorem 12 of [Conway] p. 12-13. (Contributed by Scott Fenton, 18-Jan-2026.)
((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → (𝐴 /su (2s↑s𝑁)) = ({((𝐴 -s 1s ) /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))}))
 
18-Jan-2026pw2sltdiv1d 28376 Surreal less-than relationship for division by a power of two. (Contributed by Scott Fenton, 18-Jan-2026.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 /su (2s↑s𝑁)) <s (𝐵 /su (2s↑s𝑁))))
 
18-Jan-2026ssltsnb 27733 Surreal set less-than of two singletons. (Contributed by Scott Fenton, 18-Jan-2026.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → ({𝐴} <<s {𝐵} ↔ 𝐴 <s 𝐵))
 
17-Jan-2026ex-chn2 18546 Example: sequence <" ZZ NN QQ "> is a valid chain under the equinumerosity relation in universal domain. (Contributed by Ender Ting, 17-Jan-2026.)
⟨“ℤℕℚ”⟩ ∈ ( ≈ Chain V)
 
17-Jan-2026ex-chn1 18545 Example: a doubleton of twos is a valid chain under the identity relation and domain of integers. (Contributed by Ender Ting, 17-Jan-2026.)
⟨“22”⟩ ∈ ( I Chain ℤ)
 
17-Jan-2026chnflenfi 18536 There is a finite number of chains with fixed length over finite alphabet. Trivially holds for invalid lengths as there're no matching sequences. (Contributed by Ender Ting, 5-Jan-2025.) (Revised by Ender Ting, 17-Jan-2026.)
(𝐴 ∈ Fin → {𝑎 ∈ ( < Chain 𝐴) ∣ (♯‘𝑎) = 𝑇} ∈ Fin)
 
17-Jan-2026nulchn 18527 Empty set is an increasing chain for every range and every relation. (Contributed by Ender Ting, 19-Nov-2024.) (Revised by Ender Ting, 17-Jan-2026.)
∅ ∈ ( < Chain 𝐴)
 
17-Jan-2026chnexg 18526 Chains with a set given for range form a set. (Contributed by Ender Ting, 21-Nov-2024.) (Revised by Ender Ting, 17-Jan-2026.)
(𝐴 ∈ 𝑉 → ( < Chain 𝐴) ∈ V)
 
17-Jan-2026chneq12 18522 Equality theorem for chains. (Contributed by Ender Ting, 17-Jan-2026.)
(( < = 𝑅 ∧ 𝐴 = 𝐵) → ( < Chain 𝐴) = (𝑅 Chain 𝐵))
 
17-Jan-2026chneq2 18521 Equality theorem for chains. (Contributed by Ender Ting, 17-Jan-2026.)
(𝐴 = 𝐵 → ( < Chain 𝐴) = ( < Chain 𝐵))
 
17-Jan-2026chneq1 18520 Equality theorem for chains. (Contributed by Ender Ting, 17-Jan-2026.)
( < = 𝑅 → ( < Chain 𝐴) = (𝑅 Chain 𝐴))
 
15-Jan-2026r1ssel 35139 A set is a subset of the value of the cumulative hierarchy of sets function iff it is an element of the value at the successor. (Contributed by BTernaryTau, 15-Jan-2026.)
(𝐵 ∈ On → (𝐴 ⊆ (𝑅1‘𝐵) ↔ 𝐴 ∈ (𝑅1‘suc 𝐵)))
 
15-Jan-2026fissorduni 35122 The union (supremum) of a finite set of ordinals less than a nonzero ordinal class is an element of that ordinal class. (Contributed by BTernaryTau, 15-Jan-2026.)
((𝐴 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → ∪ 𝐴 ∈ 𝐵)
 
15-Jan-2026splysubrg 33601 The symmetric polynomials form a subring of the ring of polynomials. (Contributed by Thierry Arnoux, 15-Jan-2026.)
𝑆 = (SymGrp‘𝐼)    &   𝑃 = (Base‘𝑆)    &   𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → (𝐼SymPoly𝑅) ∈ (SubRing‘(𝐼 mPoly 𝑅)))
 
15-Jan-2026mplvrpmrhm 33595 The action of permuting variables in a multivariate polynomial is a ring homomorphism. (Contributed by Thierry Arnoux, 15-Jan-2026.)
𝑆 = (SymGrp‘𝐼)    &   𝑃 = (Base‘𝑆)    &   𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   𝐹 = (𝑓 ∈ 𝑀 ↦ (𝐷𝐴𝑓))    &   𝑊 = (𝐼 mPoly 𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐷 ∈ 𝑃)    ⇒   (𝜑 → 𝐹 ∈ (𝑊 RingHom 𝑊))
 
15-Jan-2026cocnvf1o 32716 Composing with the inverse of a bijection. (Contributed by Thierry Arnoux, 15-Jan-2026.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → 𝐺:𝐴⟶𝐵)    &   (𝜑 → 𝐻:𝐴–1-1-onto→𝐴)    ⇒   (𝜑 → (𝐹 = (𝐺 ∘ 𝐻) ↔ 𝐺 = (𝐹 ∘ ◡𝐻)))
 
15-Jan-2026ofrco 32595 Function relation between function compositions. (Contributed by Thierry Arnoux, 15-Jan-2026.)
(𝜑 → 𝐹 Fn 𝐴)    &   (𝜑 → 𝐺 Fn 𝐴)    &   (𝜑 → 𝐻:𝐶⟶𝐴)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑊)    &   (𝜑 → 𝐹 ∘r 𝑅𝐺)    ⇒   (𝜑 → (𝐹 ∘ 𝐻) ∘r 𝑅(𝐺 ∘ 𝐻))
 
15-Jan-2026fnfvor 32594 Relation between two functions implies the same relation for the function value at a given 𝑋. See also fnfvof 7633. (Contributed by Thierry Arnoux, 15-Jan-2026.)
(𝜑 → 𝐹 Fn 𝐴)    &   (𝜑 → 𝐺 Fn 𝐴)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐹 ∘r 𝑅𝐺)    &   (𝜑 → 𝑋 ∈ 𝐴)    ⇒   (𝜑 → (𝐹‘𝑋)𝑅(𝐺‘𝑋))
 
15-Jan-2026elrabrd 32480 Deduction version of elrab 3643, just like elrabd 3645, but backwards direction. (Contributed by Thierry Arnoux, 15-Jan-2026.)
(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓})    ⇒   (𝜑 → 𝜒)
 
12-Jan-2026preel 38532 Predecessor is a subset of its successor. (Contributed by Peter Mazsa, 12-Jan-2026.)
(𝑁 ∈ Suc → pre 𝑁 ∈ 𝑁)
 
12-Jan-2026press 38531 Predecessor is a subset of its successor. (Contributed by Peter Mazsa, 12-Jan-2026.)
(𝑁 ∈ Suc → pre 𝑁 ⊆ 𝑁)
 
12-Jan-2026presuc 38530 pre is a left-inverse of suc. This theorem gives a clean rewrite rule that eliminates pre on explicit successors. (Contributed by Peter Mazsa, 12-Jan-2026.)
(𝑀 ∈ 𝑉 → pre suc 𝑀 = 𝑀)
 
12-Jan-2026preuniqval 38528 Uniqueness/canonicity of pre. presucmap 38527 gives one witness; this theorem gives it is the only one. It turns any predecessor proof into an equality with pre 𝑁. (Contributed by Peter Mazsa, 12-Jan-2026.)
(𝑁 ∈ ran SucMap → ∀𝑚(𝑚 SucMap 𝑁 → 𝑚 = pre 𝑁))
 
12-Jan-2026presucmap 38527 pre is really a predecessor (when it should be). This correctness theorem for pre makes it usable in proofs without unfolding ℩. This theorem gives one witness; preuniqval 38528 gives it is the only one. (Contributed by Peter Mazsa, 12-Jan-2026.)
(𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)
 
12-Jan-2026eupre2 38525 Unique predecessor exists on the range of the successor map. (Contributed by Peter Mazsa, 12-Jan-2026.)
(𝑁 ∈ 𝑉 → (𝑁 ∈ ran SucMap ↔ ∃!𝑚 𝑚 SucMap 𝑁))
 
12-Jan-2026preex 38524 The successor-predecessor exists. (Contributed by Peter Mazsa, 12-Jan-2026.)
pre 𝑁 ∈ V
 
12-Jan-2026exeupre 38523 Whenever a predecessor exists, it exists alone. (Contributed by Peter Mazsa, 12-Jan-2026.)
(𝑁 ∈ 𝑉 → (∃𝑚 𝑚 SucMap 𝑁 ↔ ∃!𝑚 𝑚 SucMap 𝑁))
 
12-Jan-2026dfpre3 38511 Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 12-Jan-2026.)
(𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚 suc 𝑚 = 𝑁))
 
12-Jan-2026dfpre2 38510 Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 12-Jan-2026.)
(𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))
 
12-Jan-2026exeupre2 38505 Whenever a predecessor exists, it exists alone. (Contributed by Peter Mazsa, 12-Jan-2026.)
(∃𝑚 suc 𝑚 = 𝑁 ↔ ∃!𝑚 suc 𝑚 = 𝑁)
 
12-Jan-2026mopre 38504 There is at most one predecessor of 𝑁. (Contributed by Peter Mazsa, 12-Jan-2026.)
∃*𝑚 suc 𝑚 = 𝑁
 
12-Jan-2026fineqvnttrclse 35165 A counterexample demonstrating that ttrclse 9624 does not hold when all sets are finite. (Contributed by BTernaryTau, 12-Jan-2026.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = suc 𝑦)}    &   𝐴 = ω    ⇒   (Fin = V → (𝑅 Se 𝐴 ∧ ¬ t++(𝑅 ↾ 𝐴) Se 𝐴))
 
12-Jan-2026fineqvnttrclselem3 35164 Lemma for fineqvnttrclse 35165. (Contributed by BTernaryTau, 12-Jan-2026.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = suc 𝑦)}    &   𝐴 = ω    &   𝐹 = (𝑣 ∈ suc suc 𝑁 ↦ ∪ {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵})    ⇒   ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵) → ∀𝑎 ∈ suc 𝑁(𝐹‘𝑎)𝑅(𝐹‘suc 𝑎))
 
12-Jan-2026fineqvnttrclselem2 35163 Lemma for fineqvnttrclse 35165. (Contributed by BTernaryTau, 12-Jan-2026.)
𝐹 = (𝑣 ∈ suc suc 𝑁 ↦ ∪ {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵})    ⇒   ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 +o (𝐹‘𝐴)) = 𝐵)
 
12-Jan-2026fineqvnttrclselem1 35162 Lemma for fineqvnttrclse 35165. (Contributed by BTernaryTau, 12-Jan-2026.)
(𝐵 ∈ (ω ∖ 1o) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
 
11-Jan-2026splyval 33600 The symmetric polynomials for a given index 𝐼 of variables and base ring 𝑅. These are the fixed points of the action 𝐴 which permutes variables. (Contributed by Thierry Arnoux, 11-Jan-2026.)
𝑆 = (SymGrp‘𝐼)    &   𝑃 = (Base‘𝑆)    &   𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    ⇒   (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴))
 
11-Jan-2026df-sply 33598 Define symmetric polynomials. See splyval 33600 for a more readable expression. (Contributed by Thierry Arnoux, 11-Jan-2026.)
SymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((Base‘(𝑖 mPoly 𝑟))FixPts(𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))))
 
11-Jan-2026mplvrpmmhm 33594 The action of permuting variables in a multivariate polynomial is a monoid homomorphism. (Contributed by Thierry Arnoux, 11-Jan-2026.)
𝑆 = (SymGrp‘𝐼)    &   𝑃 = (Base‘𝑆)    &   𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   𝐹 = (𝑓 ∈ 𝑀 ↦ (𝐷𝐴𝑓))    &   𝑊 = (𝐼 mPoly 𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐷 ∈ 𝑃)    ⇒   (𝜑 → 𝐹 ∈ (𝑊 MndHom 𝑊))
 
11-Jan-2026mplvrpmlem 33591 Lemma for mplvrpmga 33593 and others. (Contributed by Thierry Arnoux, 11-Jan-2026.)
𝑆 = (SymGrp‘𝐼)    &   𝑃 = (Base‘𝑆)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})    ⇒   (𝜑 → (𝑋 ∘ 𝐷) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
 
11-Jan-2026constcof 32606 Composition with a constant function. See also fcoconst 7073. (Contributed by Thierry Arnoux, 11-Jan-2026.)
(𝜑 → 𝐹:𝑋⟶𝐼)    &   (𝜑 → 𝑌 ∈ 𝑉)    ⇒   (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) = (𝑋 × {𝑌}))
 
10-Jan-2026finextalg 33732 A finite field extension is algebraic. Proposition 1.1 of [Lang], p. 224. (Contributed by Thierry Arnoux, 10-Jan-2026.)
(𝜑 → 𝐸/FinExt𝐹)    ⇒   (𝜑 → 𝐸/AlgExt𝐹)
 
10-Jan-2026bralgext 33731 Express the fact that a field extension 𝐸 / 𝐹 is algebraic. (Contributed by Thierry Arnoux, 10-Jan-2026.)
𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝐹)    &   (𝜑 → 𝐸 ∈ 𝑉)    &   (𝜑 → 𝐹 ∈ 𝑉)    ⇒   (𝜑 → (𝐸/AlgExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸 IntgRing 𝐶) = 𝐵)))
 
10-Jan-2026extdgfialg 33728 A finite field extension 𝐸 / 𝐹 is algebraic. Part of the proof of Proposition 1.1 of [Lang], p. 224. (Contributed by Thierry Arnoux, 10-Jan-2026.)
𝐵 = (Base‘𝐸)    &   𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹))    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐷 ∈ ℕ0)    ⇒   (𝜑 → (𝐸 IntgRing 𝐹) = 𝐵)
 
10-Jan-2026extdgfialglem2 33727 Lemma for extdgfialg 33728. (Contributed by Thierry Arnoux, 10-Jan-2026.)
𝐵 = (Base‘𝐸)    &   𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹))    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐷 ∈ ℕ0)    &   𝑍 = (0g‘𝐸)    &    · = (.r‘𝐸)    &   𝐺 = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐴:(0...𝐷)⟶𝐹)    &   (𝜑 → 𝐴 finSupp 𝑍)    &   (𝜑 → (𝐸 Σg (𝐴 ∘f · 𝐺)) = 𝑍)    &   (𝜑 → 𝐴 ≠ ((0...𝐷) × {𝑍}))    ⇒   (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹))
 
10-Jan-2026extdgfialglem1 33726 Lemma for extdgfialg 33728. (Contributed by Thierry Arnoux, 10-Jan-2026.)
𝐵 = (Base‘𝐸)    &   𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹))    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐷 ∈ ℕ0)    &   𝑍 = (0g‘𝐸)    &    · = (.r‘𝐸)    &   𝐺 = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ∃𝑎 ∈ (𝐹 ↑m (0...𝐷))(𝑎 finSupp 𝑍 ∧ ((𝐸 Σg (𝑎 ∘f · 𝐺)) = 𝑍 ∧ 𝑎 ≠ ((0...𝐷) × {𝑍}))))
 
10-Jan-2026finextfldext 33698 A finite field extension is a field extension. (Contributed by Thierry Arnoux, 10-Jan-2026.)
(𝜑 → 𝐸/FinExt𝐹)    ⇒   (𝜑 → 𝐸/FldExt𝐹)
 
10-Jan-2026srapwov 33622 The "power" operation on a subring algebra. (Contributed by Thierry Arnoux, 10-Jan-2026.)
𝐴 = ((subringAlg ‘𝑊)‘𝑆)    &   (𝜑 → 𝑊 ∈ Ring)    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (.g‘(mulGrp‘𝑊)) = (.g‘(mulGrp‘𝐴)))
 
10-Jan-2026mplvrpmga 33593 The action of permuting variables in a multivariate polynomial is a group action. (Contributed by Thierry Arnoux, 10-Jan-2026.)
𝑆 = (SymGrp‘𝐼)    &   𝑃 = (Base‘𝑆)    &   𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   (𝜑 → 𝐼 ∈ 𝑉)    ⇒   (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))
 
10-Jan-2026mplvrpmfgalem 33592 Permuting variables in a multivariate polynomial conserves finite support. (Contributed by Thierry Arnoux, 10-Jan-2026.)
𝑆 = (SymGrp‘𝐼)    &   𝑃 = (Base‘𝑆)    &   𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   (𝜑 → 𝐼 ∈ 𝑉)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝐹 ∈ 𝑀)    &   (𝜑 → 𝑄 ∈ 𝑃)    ⇒   (𝜑 → (𝑄𝐴𝐹) finSupp 0 )
 
10-Jan-2026psrbasfsupp 33579 Rewrite a finite support for nonnegative integers : For functions mapping a set 𝐼 to the nonnegative integers, having finite support can also be written as having a finite preimage of the positive integers. The latter expression is used for example in psrbas 21872, but with the former expression, theorems about finite support can be used more directly. (Contributed by Thierry Arnoux, 10-Jan-2026.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ 𝑓 finSupp 0}    ⇒   𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
 
10-Jan-2026evls1monply1 33549 Subring evaluation of a scaled monomial. (Contributed by Thierry Arnoux, 10-Jan-2026.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝑊 = (Poly1‘𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝑋 = (var1‘𝑈)    &    ↑ = (.g‘(mulGrp‘𝑊))    &    ∧ = (.g‘(mulGrp‘𝑆))    &    ∗ = ( ·𝑠 ‘𝑊)    &    · = (.r‘𝑆)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝐴 ∈ 𝑅)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑌 ∈ 𝐾)    ⇒   (𝜑 → ((𝑄‘(𝐴 ∗ (𝑁 ↑ 𝑋)))‘𝑌) = (𝐴 · (𝑁 ∧ 𝑌)))
 
10-Jan-2026fcobijfs2 32709 Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also fcobijfs 32708 and mapfien 9299. (Contributed by Thierry Arnoux, 10-Jan-2026.)
(𝜑 → 𝐺:𝑅–1-1-onto→𝑆)    &   (𝜑 → 𝑅 ∈ 𝑈)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑇 ∈ 𝑊)    &   (𝜑 → 𝑂 ∈ 𝑇)    &   𝑋 = {𝑔 ∈ (𝑇 ↑m 𝑆) ∣ 𝑔 finSupp 𝑂}    &   𝑌 = {ℎ ∈ (𝑇 ↑m 𝑅) ∣ ℎ finSupp 𝑂}    ⇒   (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝑓 ∘ 𝐺)):𝑋–1-1-onto→𝑌)
 
10-Jan-2026f1oeq3dd 32613 Equality deduction for one-to-one onto functions. (Contributed by Thierry Arnoux, 10-Jan-2026.)
(𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → 𝐹:𝐶–1-1-onto→𝐵)
 
10-Jan-2026fconst7v 32605 An alternative way to express a constant function. (Contributed by Glauco Siliprandi, 5-Feb-2022.) Removed hyphotheses as suggested by SN (Revised by Thierry Arnoux, 10-Jan-2026.)
(𝜑 → 𝐹 Fn 𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)    ⇒   (𝜑 → 𝐹 = (𝐴 × {𝐵}))
 
10-Jan-2026breq2dd 32589 Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 10-Jan-2026.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶𝑅𝐴)    ⇒   (𝜑 → 𝐶𝑅𝐵)
 
10-Jan-2026breq1dd 32588 Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 10-Jan-2026.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐴𝑅𝐶)    ⇒   (𝜑 → 𝐵𝑅𝐶)
 
8-Jan-2026sucmapleftuniq 38522 Left uniqueness of the successor mapping. (Contributed by Peter Mazsa, 8-Jan-2026.)
((𝐿 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊 ∧ 𝑁 ∈ 𝑋) → ((𝐿 SucMap 𝑁 ∧ 𝑀 SucMap 𝑁) → 𝐿 = 𝑀))
 
7-Jan-2026sucmapsuc 38521 A set is succeeded by its successor. (Contributed by Peter Mazsa, 7-Jan-2026.)
(𝑀 ∈ 𝑉 → 𝑀 SucMap suc 𝑀)
 
7-Jan-2026dmsucmap 38501 The domain of the successor map is the universe. (Contributed by Peter Mazsa, 7-Jan-2026.)
dom SucMap = V
 
7-Jan-2026relsucmap 38500 The successor map is a relation. (Contributed by Peter Mazsa, 7-Jan-2026.)
Rel SucMap
 
6-Jan-2026brsucmap 38499 Binary relation form of the successor map, general version. (Contributed by Peter Mazsa, 6-Jan-2026.)
((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 SucMap 𝑁 ↔ suc 𝑀 = 𝑁))
 
6-Jan-2026dfsuccf2 36006 Alternate definition of Scott Fenton's version of Succ, cf. df-sucmap 38495. (Contributed by Peter Mazsa, 6-Jan-2026.)
Succ = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
 
1-Jan-2026rightpos 27783 A surreal is non-negative iff all its right options are positive. (Contributed by Scott Fenton, 1-Jan-2026.)
(𝜑 → 𝐴 <<s 𝐵)    &   (𝜑 → 𝑋 = (𝐴 |s 𝐵))    ⇒   (𝜑 → ( 0s ≤s 𝑋 ↔ ∀𝑥𝑅 ∈ 𝐵 0s <s 𝑥𝑅))
 
31-Dec-2025tz9.1regs 35151 Every set has a transitive closure (the smallest transitive extension). This version of tz9.1 9626 depends on ax-regs 35145 instead of ax-reg 9485 and ax-inf2 9538. This suggests a possible answer to the third question posed in tz9.1 9626, namely that the missing property is that countably infinite classes must obey regularity. In ZF set theory we can prove this by showing that countably infinite classes are sets and thus ax-reg 9485 applies to them directly, but in a finitist context it seems that an axiom like ax-regs 35145 is required since countably infinite classes are proper classes. (Contributed by BTernaryTau, 31-Dec-2025.)
𝐴 ∈ V    ⇒   ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
 
31-Dec-2025setinds2regs 35150 Principle of set induction (or E-induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by BTernaryTau, 31-Dec-2025.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)    ⇒   𝜑
 
31-Dec-2025nelaneq 9494 A class is not an element of and equal to a class at the same time. Variant of elneq 9493 analogously to elnotel 9507 and en2lp 9503. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.) (Proof shortened by TM, 31-Dec-2025.)
¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)
 
31-Dec-2025zfregcl 9487 The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.) Avoid ax-10 2146 and ax-12 2182. (Revised by TM, 31-Dec-2025.)
(𝐴 ∈ 𝑉 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))
 
31-Dec-2025dmcosseq 5921 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-11 2162. (Revised by BTernaryTau, 23-Jun-2025.) Avoid ax-10 2146 and ax-12 2182. (Revised by TM, 31-Dec-2025.)
(ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵)
 
31-Dec-2025dmcoss 5918 Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2146 and ax-12 2182. (Revised by TM, 31-Dec-2025.)
dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
 
30-Dec-2025grlimedgnedg 48255 In general, the image of an edge of a graph by a local isomprphism is not an edge of the other graph, proven by an example (see gpg5edgnedg 48254). This theorem proves that the analogon (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ 𝐾 ∈ 𝐼)) → (𝐹 “ 𝐾) ∈ 𝐸) of grimedgi 48060 for ordinarily isomorphic graphs does not hold in general. (Contributed by AV, 30-Dec-2025.)
∃𝑔 ∈ USGraph ∃ℎ ∈ USGraph ∃𝑓 ∈ (𝑔 GraphLocIso ℎ)∃𝑎 ∈ (Vtx‘𝑔)∃𝑏 ∈ (Vtx‘𝑔)({𝑎, 𝑏} ∈ (Edg‘𝑔) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ))
 
30-Dec-2025grimedgi 48060 Graph isomorphisms map edges onto the corresponding edges. (Contributed by AV, 30-Dec-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (Edg‘𝐺)    &   𝐸 = (Edg‘𝐻)    ⇒   ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐾 ∈ 𝐼 → (𝐹 “ 𝐾) ∈ 𝐸))
 
30-Dec-2025fineqvr1ombregs 35156 All sets are finite iff all sets are hereditarily finite. (Contributed by BTernaryTau, 30-Dec-2025.)
(Fin = V ↔ ∪ (𝑅1 “ ω) = V)
 
30-Dec-2025fineqvomon 35155 If the Axiom of Infinity is negated, then the class of all natural numbers equals the proper class of all ordinal numbers. (Contributed by BTernaryTau, 30-Dec-2025.)
(Fin = V → ω = On)
 
30-Dec-2025unir1regs 35152 The cumulative hierarchy of sets covers the universe. This version of unir1 9713 replaces setind 9644 with setindregs 35149. (Contributed by BTernaryTau, 30-Dec-2025.)
∪ (𝑅1 “ On) = V
 
30-Dec-2025setindregs 35149 Set (epsilon) induction. This version of setind 9644 replaces zfregs 9629 with axregszf 35148. (Contributed by BTernaryTau, 30-Dec-2025.)
(∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
 
30-Dec-2025axregszf 35148 Derivation of zfregs 9629 using ax-regs 35145. (Contributed by BTernaryTau, 30-Dec-2025.)
(𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
 
30-Dec-2025axregscl 35147 A version of ax-regs 35145 with a class variable instead of a wff variable. Axiom D in Gödel, The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (1940), p. 6. (Contributed by BTernaryTau, 30-Dec-2025.)
(∃𝑥 𝑥 ∈ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)))
 
30-Dec-2025axreg 35146 Derivation of ax-reg 9485 from ax-regs 35145 and Tarski's FOL axiom schemes. This demonstrates the sense in which ax-regs 35145 is a stronger version of ax-reg 9485. (Contributed by BTernaryTau, 30-Dec-2025.)
(∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
 
30-Dec-2025r1omfi 35137 Hereditarily finite sets are finite sets. (Contributed by BTernaryTau, 30-Dec-2025.)
∪ (𝑅1 “ ω) ⊆ Fin
 
30-Dec-2025r1elcl 35130 Each set of the cumulative hierarchy is closed under membership. (Contributed by BTernaryTau, 30-Dec-2025.)
((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ (𝑅1‘𝐵))
 
30-Dec-2025elwf 35129 An element of a well-founded set is well-founded. (Contributed by BTernaryTau, 30-Dec-2025.)
((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ∪ (𝑅1 “ On))
 
29-Dec-2025gpg5edgnedg 48254 Two consecutive (according to the numbering) inside vertices of the Petersen graph G(5,2) are not connected by an edge, but are connected by an edge in a 5-prism G(5,1). (Contributed by AV, 29-Dec-2025.)
({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2)))
 
29-Dec-2025axregs 35166 Derivation of ax-regs 35145 from the axioms of ZF set theory. (Contributed by BTernaryTau, 29-Dec-2025.)
(∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
 
29-Dec-2025ax-regs 35145 A strong version of the Axiom of Regularity. It states that if there exists a set with property 𝜑, then there must exist a set with property 𝜑 such that none of its elements have property 𝜑. This axiom can be derived from the axioms of ZF set theory as shown in axregs 35166, but this derivation relies on ax-inf2 9538 and is thus not possible in a finitist context. (Contributed by BTernaryTau, 29-Dec-2025.)
(∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
 
29-Dec-2025optocl 5713 Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.) Shorten and reduce axiom usage. (Revised by TM, 29-Dec-2025.)
𝐷 = (𝐵 × 𝐶)    &   (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))    &   ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑)    ⇒   (𝐴 ∈ 𝐷 → 𝜓)
 
28-Dec-2025gpg5grlim 48217 A local isomorphism between the two generalized Petersen graphs G(N,K) of order 10 (𝑁 = 5), which are the Petersen graph G(5,2) and the 5-prism G(5,1). (Contributed by AV, 28-Dec-2025.)
( I ↾ ({0, 1} × (0..^5))) ∈ ((5 gPetersenGr 1) GraphLocIso (5 gPetersenGr 2))
 
28-Dec-2025clnbgr3stgrgrlim 48143 If all (closed) neighborhoods of the vertices in two simple graphs with the same order induce a subgraph which is isomorphic to an 𝑁-star, then any bijection between the vertices is a local isomorphism between the two graphs. (Contributed by AV, 28-Dec-2025.)
𝑁 ∈ ℕ0    &   𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    ⇒   (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
 
28-Dec-2025grlimgredgex 48124 Local isomorphisms between simple pseudographs map an edge onto an edge with an endpoint being the image of one of the endpoints of the first edge under the local isomorphism. (Contributed by AV, 28-Dec-2025.)
𝐼 = (Edg‘𝐺)    &   𝐸 = (Edg‘𝐻)    &   𝑉 = (Vtx‘𝐻)    &   (𝜑 → 𝐴 ∈ 𝑋)    &   (𝜑 → 𝐵 ∈ 𝑌)    &   (𝜑 → {𝐴, 𝐵} ∈ 𝐼)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))    ⇒   (𝜑 → ∃𝑣 ∈ 𝑉 {(𝐹‘𝐴), 𝑣} ∈ 𝐸)
 
28-Dec-2025grlimprclnbgrvtx 48123 For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there is a bijection 𝑓 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴), so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge between the vertices in 𝑀 containing the vertex (𝐹‘𝐴). (Contributed by AV, 28-Dec-2025.)
𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   𝐼 = (Edg‘𝐺)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   𝐽 = (Edg‘𝐻)    &   𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))
 
28-Dec-2025clnbupgreli 47959 A member of the closed neighborhood of a vertex in a pseudograph. (Contributed by AV, 28-Dec-2025.)
𝐸 = (Edg‘𝐺)    ⇒   ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ (𝐺 ClNeighbVtx 𝐾)) → (𝑁 = 𝐾 ∨ {𝑁, 𝐾} ∈ 𝐸))
 
28-Dec-2025elirrvALT 9502 Alternate proof of elirrv 9490, shorter but using more axioms. (Contributed by BTernaryTau, 28-Dec-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ 𝑥 ∈ 𝑥
 
27-Dec-2025grlimgrtrilem1 48125 Lemma 3 for grlimgrtri 48127. (Contributed by AV, 24-Aug-2025.) (Proof shortened by AV, 27-Dec-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑁 = (𝐺 ClNeighbVtx 𝑎)    &   𝐼 = (Edg‘𝐺)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    ⇒   ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑎, 𝑏} ∈ 𝐾 ∧ {𝑎, 𝑐} ∈ 𝐾 ∧ {𝑏, 𝑐} ∈ 𝐾))
 
27-Dec-2025grlimpredg 48122 For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there is a bijection 𝑓 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴), so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge in 𝐻. (Contributed by AV, 27-Dec-2025.)
𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   𝐼 = (Edg‘𝐺)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   𝐽 = (Edg‘𝐻)    &   𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽))
 
27-Dec-2025grlimprclnbgredg 48121 For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there is a bijection 𝑓 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴), so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge between the vertices in 𝑀. (Contributed by AV, 27-Dec-2025.)
𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   𝐼 = (Edg‘𝐺)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   𝐽 = (Edg‘𝐻)    &   𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿))
 
27-Dec-2025elirrv 9490 The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. This is trivial to prove from zfregfr 9501 and efrirr 5599 (see elirrvALT 9502), but this proof is direct from ax-reg 9485. (Contributed by NM, 19-Aug-1993.) Reduce axiom dependencies and make use of ax-reg 9485 directly. (Revised by BTernaryTau, 27-Dec-2025.)
¬ 𝑥 ∈ 𝑥
 
25-Dec-2025grlimprclnbgr 48120 For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there are two bijections 𝑓 and 𝑔 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴) and the edges between the vertices in 𝑁 onto the edges between the vertices in 𝑀, so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge between the vertices in 𝑀. (Contributed by AV, 25-Dec-2025.)
𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   𝐼 = (Edg‘𝐺)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   𝐽 = (Edg‘𝐻)    &   𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵})))
 
25-Dec-2025grlimedgclnbgr 48119 For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there are two bijections 𝑓 and 𝑔 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴) and the edges between the vertices in 𝑁 onto the edges between the vertices in 𝑀, so that the mapped vertices of an edge 𝐸 containing the vertex 𝐴 is an edge between the vertices in 𝑀. (Contributed by AV, 25-Dec-2025.)
𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   𝐼 = (Edg‘𝐺)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   𝐽 = (Edg‘𝐻)    &   𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ 𝐸) = (𝑔‘𝐸)))
 
25-Dec-2025clnbgrvtxedg 48118 An edge 𝐸 containing a vertex 𝐴 is an edge in the closed neighborhood of this vertex 𝐴. (Contributed by AV, 25-Dec-2025.)
𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   𝐼 = (Edg‘𝐺)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    ⇒   ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ∈ 𝐾)
 
23-Dec-2025zsoring 28333 The surreal integers form an ordered ring. Note that we have to restrict the operations here since No is a proper class. (Contributed by Scott Fenton, 23-Dec-2025.)
ℤs = (Base‘𝐾)    &   ( +s ↾ (ℤs × ℤs)) = (+g‘𝐾)    &   ( ·s ↾ (ℤs × ℤs)) = (.r‘𝐾)    &   ( ≤s ∩ (ℤs × ℤs)) = (le‘𝐾)    &    0s = (0g‘𝐾)    ⇒   𝐾 ∈ oRing
 
12-Dec-2025zs12subscl 28390 The dyadics are closed under subtraction. (Contributed by Scott Fenton, 12-Dec-2025.)
((𝐴 ∈ ℤs[1/2] ∧ 𝐵 ∈ ℤs[1/2]) → (𝐴 -s 𝐵) ∈ ℤs[1/2])
 
11-Dec-2025zs12half 28391 Half of a dyadic is a dyadic. (Contributed by Scott Fenton, 11-Dec-2025.)
(𝐴 ∈ ℤs[1/2] → (𝐴 /su 2s) ∈ ℤs[1/2])
 
11-Dec-2025zs12addscl 28388 The dyadics are closed under addition. (Contributed by Scott Fenton, 11-Dec-2025.)
((𝐴 ∈ ℤs[1/2] ∧ 𝐵 ∈ ℤs[1/2]) → (𝐴 +s 𝐵) ∈ ℤs[1/2])
 
11-Dec-2025zs12no 28387 A dyadic is a surreal. (Contributed by Scott Fenton, 11-Dec-2025.)
(𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No )
 
11-Dec-2025avgslt2d 28378 Ordering property for average. (Contributed by Scott Fenton, 11-Dec-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝐴 <s 𝐵 ↔ ((𝐴 +s 𝐵) /su 2s) <s 𝐵))
 
11-Dec-2025avgslt1d 28377 Ordering property for average. (Contributed by Scott Fenton, 11-Dec-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝐴 <s 𝐵 ↔ 𝐴 <s ((𝐴 +s 𝐵) /su 2s)))
 
11-Dec-2025pw2sltmuldiv2d 28375 Surreal less-than relationship between division and multiplication for powers of two. (Contributed by Scott Fenton, 11-Dec-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → (((2s↑s𝑁) ·s 𝐴) <s 𝐵 ↔ 𝐴 <s (𝐵 /su (2s↑s𝑁))))
 
11-Dec-2025pw2sltdivmuld 28374 Surreal less-than relationship between division and multiplication for powers of two. (Contributed by Scott Fenton, 11-Dec-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → ((𝐴 /su (2s↑s𝑁)) <s 𝐵 ↔ 𝐴 <s ((2s↑s𝑁) ·s 𝐵)))
 
11-Dec-2025pw2divscan4d 28368 Cancellation law for divison by powers of two. (Contributed by Scott Fenton, 11-Dec-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    &   (𝜑 → 𝑀 ∈ ℕ0s)    ⇒   (𝜑 → (𝐴 /su (2s↑s𝑁)) = (((2s↑s𝑀) ·s 𝐴) /su (2s↑s(𝑁 +s 𝑀))))
 
11-Dec-2025pw2divsassd 28367 An associative law for division by powers of two. (Contributed by Scott Fenton, 11-Dec-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → ((𝐴 ·s 𝐵) /su (2s↑s𝑁)) = (𝐴 ·s (𝐵 /su (2s↑s𝑁))))
 
11-Dec-2025zexpscl 28358 Closure law for surreal integer exponentiation. (Contributed by Scott Fenton, 11-Dec-2025.)
((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ ℤs)
 
11-Dec-2025nobdaymin 27717 Any non-empty class of surreals has a birthday-minimal element. (Contributed by Scott Fenton, 11-Dec-2025.)
((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ( bday ‘𝑥) = ∩ ( bday “ 𝐴))
 
10-Dec-2025sinnpoly 47015 Sine function is not a polynomial with complex coefficients. Indeed, it has infinitely many zeros but is not constant zero, contrary to fta1 26244. (Contributed by Ender Ting, 10-Dec-2025.)
¬ sin ∈ (Poly‘ℂ)
 
10-Dec-2025tannpoly 47014 The tangent function is not a polynomial with complex coefficients, as it is not defined on the whole complex plane. (Contributed by Ender Ting, 10-Dec-2025.)
¬ tan ∈ (Poly‘ℂ)
 
8-Dec-2025cjnpoly 47013 Complex conjugation operator is not a polynomial with complex coefficients. Indeed; if it was, then multiplying 𝑥 conjugate by 𝑥 itself and adding 1 would yield a nowhere-zero non-constant polynomial, contrary to the fta 27018. (Contributed by Ender Ting, 8-Dec-2025.)
¬ ∗ ∈ (Poly‘ℂ)
 
6-Dec-2025vonf1owev 35173 If 𝐹 is a bijection from the universe to the ordinals, then 𝑅 well-orders the universe. This is the ZFC version of (2 → 3) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 6-Dec-2025.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}    ⇒   (𝐹:V–1-1-onto→On → 𝑅 We V)
 
5-Dec-2025antnestALT 35759 Alternative proof of antnest 35754 from the valid schema ((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓) using laws of nested antecedents. Our proof uses only the laws antnestlaw1 35756 and antnestlaw3 35758. (Contributed by Adrian Ducourtial, 5-Dec-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)
 
5-Dec-2025antnestlaw3 35758 A law of nested antecedents. Compare with looinv 203. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
((((𝜑 → 𝜓) → 𝜒) → 𝜒) ↔ (((𝜑 → 𝜒) → 𝜓) → 𝜓))
 
5-Dec-2025antnestlaw2 35757 A law of nested antecedents. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
((((𝜑 → 𝜓) → 𝜓) → 𝜒) ↔ (((𝜑 → 𝜒) → 𝜓) → 𝜒))
 
5-Dec-2025antnestlaw1 35756 A law of nested antecedents. The converse direction is a subschema of pm2.27 42. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
((((𝜑 → 𝜓) → 𝜓) → 𝜓) ↔ (𝜑 → 𝜓))
 
5-Dec-2025antnestlaw3lem 35755 Lemma for antnestlaw3 35758. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
(¬ (((𝜑 → 𝜓) → 𝜒) → 𝜒) → ¬ (((𝜑 → 𝜒) → 𝜓) → 𝜓))
 
5-Dec-2025onvf1od 35172 If 𝐺 is a global choice function, then 𝐹 is a bijection from the ordinals to the universe. This is the ZFC version of (1 → 2) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 5-Dec-2025.)
(𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))    &   𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤}    &   𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤))    &   𝐹 = recs((𝑤 ∈ V ↦ 𝑁))    ⇒   (𝜑 → 𝐹:On–1-1-onto→V)
 
5-Dec-2025zs12zodd 28393 A dyadic fraction is either an integer or an odd number divided by a positive power of two. (Contributed by Scott Fenton, 5-Dec-2025.)
(𝐴 ∈ ℤs[1/2] → (𝐴 ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs 𝐴 = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
 
5-Dec-2025sltrecd 27764 A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 5-Dec-2025.)
(𝜑 → 𝐴 <<s 𝐵)    &   (𝜑 → 𝐶 <<s 𝐷)    &   (𝜑 → 𝑋 = (𝐴 |s 𝐵))    &   (𝜑 → 𝑌 = (𝐶 |s 𝐷))    ⇒   (𝜑 → (𝑋 <s 𝑌 ↔ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)))
 
5-Dec-2025slerecd 27762 A comparison law for surreals considered as cuts of sets of surreals. Definition from [Conway] p. 4. Theorem 4 of [Alling] p. 186. Theorem 2.5 of [Gonshor] p. 9. (Contributed by Scott Fenton, 5-Dec-2025.)
(𝜑 → 𝐴 <<s 𝐵)    &   (𝜑 → 𝐶 <<s 𝐷)    &   (𝜑 → 𝑋 = (𝐴 |s 𝐵))    &   (𝜑 → 𝑌 = (𝐶 |s 𝐷))    ⇒   (𝜑 → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)))
 
4-Dec-2025onvf1odlem4 35171 Lemma for onvf1od 35172. If the range of 𝐹 does not exist, then it must equal the universe. (Contributed by BTernaryTau, 4-Dec-2025.)
(𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))    &   𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤}    &   𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤))    &   𝐹 = recs((𝑤 ∈ V ↦ 𝑁))    &   𝐵 = ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}    &   𝐶 = (𝐺‘((𝑅1‘𝐵) ∖ (𝐹 “ 𝑡)))    ⇒   (𝜑 → (¬ ran 𝐹 ∈ V → ran 𝐹 = V))
 
2-Dec-2025onvf1odlem3 35170 Lemma for onvf1od 35172. The value of 𝐹 at an ordinal 𝐴. (Contributed by BTernaryTau, 2-Dec-2025.)
𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤}    &   𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤))    &   𝐹 = recs((𝑤 ∈ V ↦ 𝑁))    &   𝐵 = ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)}    &   𝐶 = (𝐺‘((𝑅1‘𝐵) ∖ (𝐹 “ 𝐴)))    ⇒   (𝐴 ∈ On → (𝐹‘𝐴) = 𝐶)
 
2-Dec-2025onvf1odlem2 35169 Lemma for onvf1od 35172. (Contributed by BTernaryTau, 2-Dec-2025.)
(𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))    &   𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}    &   𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ 𝐴))    ⇒   (𝜑 → (𝐴 ∈ 𝑉 → 𝑁 ∈ ((𝑅1‘𝑀) ∖ 𝐴)))
 
2-Dec-2025onvf1odlem1 35168 Lemma for onvf1od 35172. (Contributed by BTernaryTau, 2-Dec-2025.)
(𝐴 ∈ 𝑉 → ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
 
1-Dec-2025sn-msqgt0d 42604 A nonzero square is positive. (Contributed by SN, 1-Dec-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 ≠ 0)    ⇒   (𝜑 → 0 < (𝐴 · 𝐴))
 
1-Dec-2025sn-mullt0d 42603 The product of two negative numbers is positive. (Contributed by SN, 1-Dec-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 < 0)    &   (𝜑 → 𝐵 < 0)    ⇒   (𝜑 → 0 < (𝐴 · 𝐵))
 
1-Dec-2025elabgt 3623 Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3628.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom usage. (Revised by GG, 12-Oct-2024.) (Proof shortened by Wolf Lammen, 11-May-2025.) (Proof shortened by SN, 1-Dec-2025.)
((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
 
30-Nov-2025eluz3nn 12789 An integer greater than or equal to 3 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.) (Proof shortened by AV, 30-Nov-2025.)
(𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
 
28-Nov-2025eqscut3 27766 A variant of the simplicity theorem - if 𝐵 lies between the cut sets of 𝐴 but none of its options do, then 𝐴 = 𝐵. Theorem 11 of [Conway] p. 23. (Contributed by Scott Fenton, 28-Nov-2025.)
(𝜑 → 𝐿 <<s 𝑅)    &   (𝜑 → 𝑀 <<s 𝑆)    &   (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   (𝜑 → 𝐵 = (𝑀 |s 𝑆))    &   (𝜑 → 𝐿 <<s {𝐵})    &   (𝜑 → {𝐵} <<s 𝑅)    &   (𝜑 → ∀𝑥𝑂 ∈ (𝑀 ∪ 𝑆) ¬ (𝐿 <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s 𝑅))    ⇒   (𝜑 → 𝐴 = 𝐵)
 
27-Nov-2025difmodm1lt 47483 The difference between an integer modulo a positive integer and the integer decreased by 1 modulo the same modulus is less than the modulus decreased by 1 (if the modulus is greater than 2). This theorem would not be valid for an odd 𝐴 and 𝑁 = 2, since ((𝐴 mod 𝑁) − ((𝐴 − 1) mod 𝑁)) would be (1 − 0) = 1 which is not less than (𝑁 − 1) = 1. (Contributed by AV, 6-Jun-2012.) (Proof shortened by SN, 27-Nov-2025.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 2 < 𝑁) → ((𝐴 mod 𝑁) − ((𝐴 − 1) mod 𝑁)) < (𝑁 − 1))
 
26-Nov-2025cmdlan 49797 To each colimit of a diagram there is a corresponding left Kan extention of the diagram along a functor to a terminal category. The morphism parts coincide, while the object parts are one-to-one correspondent (diag1f1o 49659). (Contributed by Zhi Wang, 26-Nov-2025.)
(𝜑 → 1 ∈ TermCat)    &   (𝜑 → 𝐺 ∈ (𝐷 Func 1 ))    &   𝐿 = (𝐶Δfunc 1 )    &   (𝜑 → 𝑌 = ((1st ‘𝐿)‘𝑋))    ⇒   (𝜑 → (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Lan 𝐶)𝐹)𝑀))
 
26-Nov-2025lmdran 49796 To each limit of a diagram there is a corresponding right Kan extention of the diagram along a functor to a terminal category. The morphism parts coincide, while the object parts are one-to-one correspondent (diag1f1o 49659). (Contributed by Zhi Wang, 26-Nov-2025.)
(𝜑 → 1 ∈ TermCat)    &   (𝜑 → 𝐺 ∈ (𝐷 Func 1 ))    &   𝐿 = (𝐶Δfunc 1 )    &   (𝜑 → 𝑌 = ((1st ‘𝐿)‘𝑋))    ⇒   (𝜑 → (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Ran 𝐶)𝐹)𝑀))
 
26-Nov-2025ranval3 49756 The set of right Kan extensions is the set of universal pairs. (Contributed by Zhi Wang, 26-Nov-2025.)
𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))    &   𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))    &   𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)    ⇒   (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋))
 
26-Nov-2025ffthoppf 49290 The opposite functor of a fully faithful functor is also full and faithful. (Contributed by Zhi Wang, 26-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))    ⇒   (𝜑 → ( oppFunc ‘𝐹) ∈ ((𝑂 Full 𝑃) ∩ (𝑂 Faith 𝑃)))
 
26-Nov-2025fthoppf 49289 The opposite functor of a faithful functor is also faithful. Proposition 3.43(c) in [Adamek] p. 39. (Contributed by Zhi Wang, 26-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐹 ∈ (𝐶 Faith 𝐷))    ⇒   (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Faith 𝑃))
 
26-Nov-2025fulloppf 49288 The opposite functor of a full functor is also full. Proposition 3.43(d) in [Adamek] p. 39. (Contributed by Zhi Wang, 26-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐹 ∈ (𝐶 Full 𝐷))    ⇒   (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Full 𝑃))
 
26-Nov-2025cofuoppf 49275 Composition of opposite functors. (Contributed by Zhi Wang, 26-Nov-2025.)
(𝜑 → (𝐺 ∘func 𝐹) = 𝐾)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    ⇒   (𝜑 → (( oppFunc ‘𝐺) ∘func ( oppFunc ‘𝐹)) = ( oppFunc ‘𝐾))
 
26-Nov-2025mullt0b2d 42602 When the second term is negative, the first term is positive iff the product is negative. (Contributed by SN, 26-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐵 < 0)    ⇒   (𝜑 → (0 < 𝐴 ↔ (𝐴 · 𝐵) < 0))
 
26-Nov-2025mullt0b1d 42601 When the first term is negative, the second term is positive iff the product is negative. (Contributed by SN, 26-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 < 0)    ⇒   (𝜑 → (0 < 𝐵 ↔ (𝐴 · 𝐵) < 0))
 
26-Nov-2025mulltgt0d 42600 Negative times positive is negative. (Contributed by SN, 26-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 < 0)    &   (𝜑 → 0 < 𝐵)    ⇒   (𝜑 → (𝐴 · 𝐵) < 0)
 
26-Nov-2025sn-reclt0d 42599 The reciprocal of a negative real is negative. (Contributed by SN, 26-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 0)    ⇒   (𝜑 → (1 /ℝ 𝐴) < 0)
 
26-Nov-2025sn-recgt0d 42595 The reciprocal of a positive real is positive. (Contributed by SN, 26-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    ⇒   (𝜑 → 0 < (1 /ℝ 𝐴))
 
25-Nov-2025prcofdiag 49519 A diagonal functor post-composed by a pre-composition functor is another diagonal functor. (Contributed by Zhi Wang, 25-Nov-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   𝑀 = (𝐶Δfunc𝐸)    &   (𝜑 → 𝐹 ∈ (𝐸 Func 𝐷))    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → (⟨𝐷, 𝐶⟩ −∘F 𝐹) = 𝐺)    ⇒   (𝜑 → (𝐺 ∘func 𝐿) = 𝑀)
 
25-Nov-2025prcofdiag1 49518 A constant functor pre-composed by a functor is another constant functor. (Contributed by Zhi Wang, 25-Nov-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   𝑀 = (𝐶Δfunc𝐸)    &   (𝜑 → 𝐹 ∈ (𝐸 Func 𝐷))    &   (𝜑 → 𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) = ((1st ‘𝑀)‘𝑋))
 
25-Nov-2025uptr2a 49347 Universal property and fully faithful functor surjective on objects. (Contributed by Zhi Wang, 25-Nov-2025.)
𝐴 = (Base‘𝐶)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑌 = ((1st ‘𝐾)‘𝑋))    &   (𝜑 → (𝐺 ∘func 𝐾) = 𝐹)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐾 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))    &   (𝜑 → (1st ‘𝐾):𝐴–onto→𝐵)    ⇒   (𝜑 → (𝑋(𝐹(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑌(𝐺(𝐷 UP 𝐸)𝑍)𝑀))
 
25-Nov-2025uptr2 49346 Universal property and fully faithful functor surjective on objects. (Contributed by Zhi Wang, 25-Nov-2025.)
𝐴 = (Base‘𝐶)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑌 = (𝑅‘𝑋))    &   (𝜑 → 𝑅:𝐴–onto→𝐵)    &   (𝜑 → 𝑅((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝑆)    &   (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝑅, 𝑆⟩) = ⟨𝐹, 𝐺⟩)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    ⇒   (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀))
 
25-Nov-2025xpco2 48981 Composition of a Cartesian product with a function. (Contributed by Zhi Wang, 25-Nov-2025.)
(𝐹:𝐴⟶𝐵 → ((𝐵 × 𝐶) ∘ 𝐹) = (𝐴 × 𝐶))
 
25-Nov-2025ffvbr 48980 Relation with function value. (Contributed by Zhi Wang, 25-Nov-2025.)
((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋𝐹(𝐹‘𝑋))
 
25-Nov-2025rerecid2 42568 Multiplication of a number and its reciprocal. (Contributed by SN, 25-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 ≠ 0)    ⇒   (𝜑 → ((1 /ℝ 𝐴) · 𝐴) = 1)
 
25-Nov-2025rerecid 42567 Multiplication of a number and its reciprocal. (Contributed by SN, 25-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 ≠ 0)    ⇒   (𝜑 → (𝐴 · (1 /ℝ 𝐴)) = 1)
 
25-Nov-2025sn-rereccld 42566 Closure law for reciprocal. (Contributed by SN, 25-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 ≠ 0)    ⇒   (𝜑 → (1 /ℝ 𝐴) ∈ ℝ)
 
25-Nov-2025redivcan3d 42565 A cancellation law for division. (Contributed by SN, 25-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐵 ≠ 0)    ⇒   (𝜑 → ((𝐵 · 𝐴) /ℝ 𝐵) = 𝐴)
 
25-Nov-2025redivcan2d 42564 A cancellation law for division. (Contributed by SN, 25-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐵 ≠ 0)    ⇒   (𝜑 → (𝐵 · (𝐴 /ℝ 𝐵)) = 𝐴)
 
25-Nov-2025redivmuld 42563 Relationship between division and multiplication. (Contributed by SN, 25-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐶 ∈ ℝ)    &   (𝜑 → 𝐶 ≠ 0)    ⇒   (𝜑 → ((𝐴 /ℝ 𝐶) = 𝐵 ↔ (𝐶 · 𝐵) = 𝐴))
 
25-Nov-2025sn-redivcld 42562 Closure law for real division. (Contributed by SN, 25-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐵 ≠ 0)    ⇒   (𝜑 → (𝐴 /ℝ 𝐵) ∈ ℝ)
 
25-Nov-2025rediveud 42561 Existential uniqueness of real quotients. (Contributed by SN, 25-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐵 ≠ 0)    ⇒   (𝜑 → ∃!𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴)
 
25-Nov-2025redivvald 42560 Value of real division, which is the (unique) real 𝑥 such that (𝐵 · 𝑥) = 𝐴. (Contributed by SN, 25-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐵 ≠ 0)    ⇒   (𝜑 → (𝐴 /ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴))
 
25-Nov-2025df-rediv 42559 Define division between real numbers. This operator saves ax-mulcom 11077 over df-div 11782 in certain situations. (Contributed by SN, 25-Nov-2025.)
/ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))
 
25-Nov-2025uniqsw 8705 The union of a quotient set. More restrictive antecedent; kept for backward compatibility; for new work, prefer uniqs 8704. (Contributed by NM, 9-Dec-2008.) (Proof shortened by AV, 25-Nov-2025.)
(𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))
 
25-Nov-2025ecelqsw 8699 Membership of an equivalence class in a quotient set. More restrictive antecedent; kept for backward compatibility; for new work, prefer ecelqs 8698. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.) (Proof shortened by AV, 25-Nov-2025.)
((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
 
24-Nov-2025f1omo 49017 There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 49016 assuming ax-un 7674 (see f1omoALT 49019). (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by SN, 24-Nov-2025.)
(𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
 
24-Nov-2025mulgt0b2d 42596 Biconditional, deductive form of mulgt0 11197. The first factor is positive iff the product is. (Contributed by SN, 24-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐵)    ⇒   (𝜑 → (0 < 𝐴 ↔ 0 < (𝐴 · 𝐵)))
 
24-Nov-2025sn-remul0ord 42526 A product is zero iff one of its factors are zero. (Contributed by SN, 24-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    ⇒   (𝜑 → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0)))
 
23-Nov-2025lgricngricex 48253 There are two different locally isomorphic graphs which are not isomorphic. (Contributed by AV, 23-Nov-2025.)
∃𝑔∃ℎ(𝑔 ≃𝑙𝑔𝑟 ℎ ∧ ¬ 𝑔 ≃𝑔𝑟 ℎ)
 
23-Nov-2025dmqsblocks 38971 If the pet 38969 span (𝑅 ⋉ (◡ E ↾ 𝐴)) partitions 𝐴, then every block 𝑢 ∈ 𝐴 is of the form [𝑣] for some 𝑣 that not only lies in the domain but also has at least one internal element 𝑐 and at least one 𝑅-target 𝑏 (cf. also the comments of qseq 38766). It makes explicit that pet 38969 gives active representatives for each block, without ever forcing 𝑣 = 𝑢. (Contributed by Peter Mazsa, 23-Nov-2025.)
((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
 
23-Nov-2025eceldmqsxrncnvepres2 38481 An (𝑅 ⋉ (◡ E ↾ 𝐴))-coset in its domain quotient. In the pet 38969 span (𝑅 ⋉ (◡ E ↾ 𝐴)), a block [ B ] lies in the domain quotient exactly when its representative 𝐵 belongs to 𝐴 and actually fires at least one arrow (has some 𝑥 ∈ 𝐵 and some 𝑦 with 𝐵𝑅𝑦). (Contributed by Peter Mazsa, 23-Nov-2025.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋) → ([𝐵](𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝐵 ∧ ∃𝑦 𝐵𝑅𝑦)))
 
23-Nov-2025eceldmqsxrncnvepres 38480 An (𝑅 ⋉ (◡ E ↾ 𝐴))-coset in its domain quotient. (Contributed by Peter Mazsa, 23-Nov-2025.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋) → ([𝐵](𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅ ∧ [𝐵]𝑅 ≠ ∅)))
 
23-Nov-2025eldmxrncnvepres2 38479 Element of the domain of the range product with restricted converse epsilon relation. This identifies the domain of the pet 38969 span (𝑅 ⋉ (◡ E ↾ 𝐴)): a 𝐵 belongs to the domain of the span exactly when 𝐵 is in 𝐴 and has at least one 𝑥 ∈ 𝐵 and 𝑦 with 𝐵𝑅𝑦. (Contributed by Peter Mazsa, 23-Nov-2025.)
(𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝐵 ∧ ∃𝑦 𝐵𝑅𝑦)))
 
23-Nov-2025eldmxrncnvepres 38478 Element of the domain of the range product with restricted converse epsilon relation. (Contributed by Peter Mazsa, 23-Nov-2025.)
(𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅ ∧ [𝐵]𝑅 ≠ ∅)))
 
23-Nov-2025dmxrncnvepres 38476 Domain of the range product with restricted converse epsilon relation. (Contributed by Peter Mazsa, 23-Nov-2025.)
dom (𝑅 ⋉ (◡ E ↾ 𝐴)) = (dom (𝑅 ↾ 𝐴) ∖ {∅})
 
23-Nov-2025dmxrncnvep 38433 Domain of the range product with converse epsilon relation. (Contributed by Peter Mazsa, 23-Nov-2025.)
dom (𝑅 ⋉ ◡ E ) = (dom 𝑅 ∖ {∅})
 
23-Nov-2025dmcnvep 38432 Domain of converse epsilon relation. (Contributed by Peter Mazsa, 30-Jan-2018.) (Revised by Peter Mazsa, 23-Nov-2025.)
dom ◡ E = (V ∖ {∅})
 
23-Nov-2025eldmres3 38335 Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 23-Nov-2025.)
(𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ [𝐵]𝑅 ≠ ∅)))
 
22-Nov-2025gpg5ngric 48252 The two generalized Petersen graphs G(5,K) of order 10, which are the Petersen graph G(5,2) and the 5-prism G(5,1), are not isomorphic. (Contributed by AV, 22-Nov-2025.)
¬ (5 gPetersenGr 1) ≃𝑔𝑟 (5 gPetersenGr 2)
 
22-Nov-2025pg4cyclnex 48251 In the Petersen graph G(5,2), there is no cycle of length 4. (Contributed by AV, 22-Nov-2025.)
¬ ∃𝑝∃𝑓(𝑓(Cycles‘(5 gPetersenGr 2))𝑝 ∧ (♯‘𝑓) = 4)
 
22-Nov-2025gpg5grlic 48218 The two generalized Petersen graphs G(N,K) of order 10 (𝑁 = 5), which are the Petersen graph G(5,2) and the 5-prism G(5,1), are locally isomorphic. (Contributed by AV, 29-Sep-2025.) (Proof shortened by AV, 22-Nov-2025.)
(5 gPetersenGr 1) ≃𝑙𝑔𝑟 (5 gPetersenGr 2)
 
22-Nov-2025gpg3nbgrvtx1 48202 In a generalized Petersen graph 𝐺, every inside vertex has exactly three (different) neighbors. (Contributed by AV, 3-Sep-2025.) (Proof shortened by AV, 22-Nov-2025.)
𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   𝐺 = (𝑁 gPetersenGr 𝐾)    &   𝑉 = (Vtx‘𝐺)    &   𝑈 = (𝐺 NeighbVtx 𝑋)    ⇒   (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3)
 
22-Nov-2025modm1nem2 47493 A nonnegative integer less than a modulus greater than 4 minus one/minus two are not equal modulo the modulus. (Contributed by AV, 22-Nov-2025.)
𝐼 = (0..^𝑁)    ⇒   ((𝑁 ∈ (ℤ≥‘5) ∧ 𝑌 ∈ 𝐼) → ((𝑌 − 1) mod 𝑁) ≠ ((𝑌 − 2) mod 𝑁))
 
22-Nov-2025modm1nep2 47492 A nonnegative integer less than a modulus greater than 4 plus one/minus two are not equal modulo the modulus. (Contributed by AV, 22-Nov-2025.)
𝐼 = (0..^𝑁)    ⇒   ((𝑁 ∈ (ℤ≥‘5) ∧ 𝑌 ∈ 𝐼) → ((𝑌 − 1) mod 𝑁) ≠ ((𝑌 + 2) mod 𝑁))
 
22-Nov-2025modp2nep1 47491 A nonnegative integer less than a modulus greater than 4 plus one/plus two are not equal modulo the modulus. (Contributed by AV, 22-Nov-2025.)
𝐼 = (0..^𝑁)    ⇒   ((𝑁 ∈ (ℤ≥‘5) ∧ 𝑌 ∈ 𝐼) → ((𝑌 + 2) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁))
 
22-Nov-2025modm2nep1 47490 A nonnegative integer less than a modulus greater than 4 plus one/minus two are not equal modulo the modulus. (Contributed by AV, 22-Nov-2025.)
𝐼 = (0..^𝑁)    ⇒   ((𝑁 ∈ (ℤ≥‘5) ∧ 𝑌 ∈ 𝐼) → ((𝑌 − 2) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁))
 
22-Nov-2025dmxrn 38431 Domain of the range product. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 22-Nov-2025.)
dom (𝑅 ⋉ 𝑆) = (dom 𝑅 ∩ dom 𝑆)
 
22-Nov-2025brxrncnvep 38430 The range product with converse epsilon relation. (Contributed by Peter Mazsa, 22-Jun-2020.) (Revised by Peter Mazsa, 22-Nov-2025.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(𝑅 ⋉ ◡ E )⟨𝐵, 𝐶⟩ ↔ (𝐶 ∈ 𝐴 ∧ 𝐴𝑅𝐵)))
 
22-Nov-2025bdayle 27862 A condition for bounding a birthday above. (Contributed by Scott Fenton, 22-Nov-2025.)
((𝑋 ∈ No ∧ Ord 𝑂) → (( bday ‘𝑋) ⊆ 𝑂 ↔ ∀𝑦 ∈ ( O ‘( bday ‘𝑋))( bday ‘𝑦) ∈ 𝑂))
 
22-Nov-2025bdayiun 27861 The birthday of a surreal is the least upper bound of the successors of the birthdays of its options. This is the definition of the birthday of a combinatorial game in the Lean Combinatorial Game Theory library at https://github.com/vihdzp/combinatorial-games. (Contributed by Scott Fenton, 22-Nov-2025.)
(𝐴 ∈ No → ( bday ‘𝐴) = ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
 
22-Nov-2025nn0absidi 15340 A nonnegative integer is its own absolute value (inference form). (Contributed by AV, 22-Nov-2025.)
𝑁 ∈ ℕ0    ⇒   (abs‘𝑁) = 𝑁
 
22-Nov-2025nn0absid 15339 A nonnegative integer is its own absolute value. (Contributed by AV, 22-Nov-2025.)
(𝑁 ∈ ℕ0 → (abs‘𝑁) = 𝑁)
 
22-Nov-2025eluz5nn 12791 An integer greater than or equal to 5 is a positive integer. (Contributed by AV, 22-Nov-2025.)
(𝑁 ∈ (ℤ≥‘5) → 𝑁 ∈ ℕ)
 
22-Nov-2025eceldmqs 8717 𝑅-coset in its domain quotient. This is the bridge between 𝐴 in the domain and its block [𝐴]𝑅 in its domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.) (Revised by Peter Mazsa, 22-Nov-2025.)
(𝑅 ∈ 𝑉 → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))
 
22-Nov-2025ecelqsdmb 8716 𝑅-coset of 𝐵 in a quotient set, biconditional version. (Contributed by Peter Mazsa, 17-Apr-2019.) (Revised by Peter Mazsa, 22-Nov-2025.)
(((𝑅 ↾ 𝐴) ∈ 𝑉 ∧ dom 𝑅 = 𝐴) → ([𝐵]𝑅 ∈ (𝐴 / 𝑅) ↔ 𝐵 ∈ 𝐴))
 
22-Nov-2025ecelqs 8698 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 22-Nov-2025.)
(((𝑅 ↾ 𝐴) ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
 
21-Nov-2025ranpropd 49741 If the categories have the same set of objects, morphisms, and compositions, then they have the same right Kan extensions. (Contributed by Zhi Wang, 21-Nov-2025.)
(𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   (𝜑 → (Homf ‘𝐸) = (Homf ‘𝐹))    &   (𝜑 → (compf‘𝐸) = (compf‘𝐹))    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐸 ∈ 𝑉)    &   (𝜑 → 𝐹 ∈ 𝑉)    ⇒   (𝜑 → (⟨𝐴, 𝐶⟩ Ran 𝐸) = (⟨𝐵, 𝐷⟩ Ran 𝐹))
 
21-Nov-2025lanpropd 49740 If the categories have the same set of objects, morphisms, and compositions, then they have the same left Kan extensions. (Contributed by Zhi Wang, 21-Nov-2025.)
(𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   (𝜑 → (Homf ‘𝐸) = (Homf ‘𝐹))    &   (𝜑 → (compf‘𝐸) = (compf‘𝐹))    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐸 ∈ 𝑉)    &   (𝜑 → 𝐹 ∈ 𝑉)    ⇒   (𝜑 → (⟨𝐴, 𝐶⟩ Lan 𝐸) = (⟨𝐵, 𝐷⟩ Lan 𝐹))
 
21-Nov-2025prcofpropd 49504 If the categories have the same set of objects, morphisms, and compositions, then they have the same pre-composition functors. (Contributed by Zhi Wang, 21-Nov-2025.)
(𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐹 ∈ 𝑊)    ⇒   (𝜑 → (⟨𝐴, 𝐶⟩ −∘F 𝐹) = (⟨𝐵, 𝐷⟩ −∘F 𝐹))
 
21-Nov-2025pgnbgreunbgrlem5 48247 Lemma 5 for pgnbgreunbgr 48249. Impossible cases. (Contributed by AV, 21-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ((𝐿 = ⟨0, (((2nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2nd ‘𝑋) − 1) mod 5)⟩) → ((𝐾 = ⟨0, (((2nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2nd ‘𝑋) − 1) mod 5)⟩) → ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
 
21-Nov-2025pgnbgreunbgrlem5lem1 48244 Lemma 1 for pgnbgreunbgrlem5 48247. (Contributed by AV, 21-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ((((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨1, 𝑏⟩} ∈ 𝐸) → ¬ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸)
 
21-Nov-2025pgnioedg5 48236 An inside and an outside vertex not adjacent in a Petersen graph. (Contributed by AV, 21-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑦 ∈ (0..^5) → ¬ {⟨1, ((𝑦 − 1) mod 5)⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸)
 
21-Nov-2025pgnioedg4 48235 An inside and an outside vertex not adjacent in a Petersen graph. (Contributed by AV, 21-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑦 ∈ (0..^5) → ¬ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸)
 
21-Nov-2025pgnioedg3 48234 An inside and an outside vertex not adjacent in a Petersen graph. (Contributed by AV, 21-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑦 ∈ (0..^5) → ¬ {⟨1, ((𝑦 + 2) mod 5)⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸)
 
21-Nov-2025pgnioedg2 48233 An inside and an outside vertex not adjacent in a Petersen graph. (Contributed by AV, 21-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑦 ∈ (0..^5) → ¬ {⟨1, ((𝑦 + 2) mod 5)⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸)
 
21-Nov-2025pgnioedg1 48232 An inside and an outside vertex not adjacent in a Petersen graph. (Contributed by AV, 21-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑦 ∈ (0..^5) → ¬ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸)
 
21-Nov-2025modlt0b 47487 An integer with an absolute value less than a positive integer is 0 modulo the positive integer iff it is 0. (Contributed by AV, 21-Nov-2025.)
((𝑁 ∈ ℕ ∧ 𝑋 ∈ ℤ ∧ (abs‘𝑋) < 𝑁) → ((𝑋 mod 𝑁) = 0 ↔ 𝑋 = 0))
 
21-Nov-2025zabs0b 15223 An integer has an absolute value less than 1 iff it is 0. (Contributed by AV, 21-Nov-2025.)
(𝑋 ∈ ℤ → ((abs‘𝑋) < 1 ↔ 𝑋 = 0))
 
20-Nov-2025termolmd 49795 Terminal objects are the object part of limits of the empty diagram. (Contributed by Zhi Wang, 20-Nov-2025.)
(TermO‘𝐶) = dom (∅(𝐶 Limit ∅)∅)
 
20-Nov-2025cmddu 49793 The duality of limits and colimits: colimits of a diagram are limits of an opposite diagram in opposite categories. (Contributed by Zhi Wang, 20-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝐺 = ( oppFunc ‘𝐹)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    ⇒   (𝜑 → ((𝐶 Colimit 𝐷)‘𝐹) = ((𝑂 Limit 𝑃)‘𝐺))
 
20-Nov-2025lmddu 49792 The duality of limits and colimits: limits of a diagram are colimits of an opposite diagram in opposite categories. (Contributed by Zhi Wang, 20-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝐺 = ( oppFunc ‘𝐹)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    ⇒   (𝜑 → ((𝐶 Limit 𝐷)‘𝐹) = ((𝑂 Colimit 𝑃)‘𝐺))
 
20-Nov-2025cmdpropd 49783 If the categories have the same set of objects, morphisms, and compositions, then they have the same colimits. (Contributed by Zhi Wang, 20-Nov-2025.)
(𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑉)    ⇒   (𝜑 → (𝐴 Colimit 𝐶) = (𝐵 Colimit 𝐷))
 
20-Nov-2025lmdpropd 49782 If the categories have the same set of objects, morphisms, and compositions, then they have the same limits. (Contributed by Zhi Wang, 20-Nov-2025.)
(𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑉)    ⇒   (𝜑 → (𝐴 Limit 𝐶) = (𝐵 Limit 𝐷))
 
20-Nov-2025cmdrcl 49777 Reverse closure for a colimit of a diagram. (Contributed by Zhi Wang, 20-Nov-2025.)
(𝑋 ∈ ((𝐶 Colimit 𝐷)‘𝐹) → 𝐹 ∈ (𝐷 Func 𝐶))
 
20-Nov-2025lmdrcl 49776 Reverse closure for a limit of a diagram. (Contributed by Zhi Wang, 20-Nov-2025.)
(𝑋 ∈ ((𝐶 Limit 𝐷)‘𝐹) → 𝐹 ∈ (𝐷 Func 𝐶))
 
20-Nov-2025diagpropd 49417 If two categories have the same set of objects, morphisms, and compositions, then they have same diagonal functors. (Contributed by Zhi Wang, 20-Nov-2025.)
(𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   (𝜑 → 𝐴 ∈ Cat)    &   (𝜑 → 𝐵 ∈ Cat)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    ⇒   (𝜑 → (𝐴Δfunc𝐶) = (𝐵Δfunc𝐷))
 
20-Nov-20252ndfpropd 49416 If two categories have the same set of objects, morphisms, and compositions, then they have same second projection functors. (Contributed by Zhi Wang, 20-Nov-2025.)
(𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   (𝜑 → 𝐴 ∈ Cat)    &   (𝜑 → 𝐵 ∈ Cat)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    ⇒   (𝜑 → (𝐴 2ndF 𝐶) = (𝐵 2ndF 𝐷))
 
20-Nov-20251stfpropd 49415 If two categories have the same set of objects, morphisms, and compositions, then they have same first projection functors. (Contributed by Zhi Wang, 20-Nov-2025.)
(𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   (𝜑 → 𝐴 ∈ Cat)    &   (𝜑 → 𝐵 ∈ Cat)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    ⇒   (𝜑 → (𝐴 1stF 𝐶) = (𝐵 1stF 𝐷))
 
20-Nov-2025uppropd 49306 If two categories have the same set of objects, morphisms, and compositions, then they have the same universal pairs. (Contributed by Zhi Wang, 20-Nov-2025.)
(𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑉)    ⇒   (𝜑 → (𝐴 UP 𝐶) = (𝐵 UP 𝐷))
 
20-Nov-2025reueqbidva 48930 Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reueqbidv 3385. (Contributed by Zhi Wang, 20-Nov-2025.)
(𝜑 → 𝐴 = 𝐵)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒))
 
20-Nov-2025pgnbgreunbgrlem6 48248 Lemma 6 for pgnbgreunbgr 48249. (Contributed by AV, 20-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))
 
20-Nov-2025pgnbgreunbgrlem5lem3 48246 Lemma 3 for pgnbgreunbgrlem5 48247. (Contributed by AV, 20-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ((((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨1, 𝑏⟩} ∈ 𝐸) → ¬ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸)
 
20-Nov-2025pgnbgreunbgrlem5lem2 48245 Lemma 2 for pgnbgreunbgrlem5 48247. (Contributed by AV, 20-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ((((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨1, 𝑏⟩} ∈ 𝐸) → ¬ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸)
 
20-Nov-2025pgnbgreunbgrlem4 48243 Lemma 4 for pgnbgreunbgr 48249. (Contributed by AV, 20-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ((𝐿 = ⟨1, (((2nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝐿 = ⟨1, (((2nd ‘𝑋) − 2) mod 5)⟩) → ((𝐾 = ⟨1, (((2nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝐾 = ⟨1, (((2nd ‘𝑋) − 2) mod 5)⟩) → ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
 
20-Nov-2025gpgedg2iv 48191 The edges of the generalized Petersen graph GPG(N,K) between two inside vertices. (Contributed by AV, 20-Nov-2025.)
𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   𝐼 = (0..^𝑁)    &   𝐺 = (𝑁 gPetersenGr 𝐾)    &   𝐸 = (Edg‘𝐺)    ⇒   ((𝑁 ∈ (ℤ≥‘5) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) ∧ (𝐾 ∈ 𝐽 ∧ ((4 · 𝐾) mod 𝑁) ≠ 0)) → (({⟨1, ((𝑌 − 𝐾) mod 𝑁)⟩, ⟨1, 𝑋⟩} ∈ 𝐸 ∧ {⟨1, 𝑋⟩, ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩} ∈ 𝐸) ↔ 𝑋 = 𝑌))
 
20-Nov-20258mod5e3 47484 8 modulo 5 is 3. (Contributed by AV, 20-Nov-2025.)
(8 mod 5) = 3
 
19-Nov-2025oppfdiag 49541 A diagonal functor for opposite categories is the opposite functor of the diagonal functor for original categories post-composed by an isomorphism (fucoppc 49535). (Contributed by Zhi Wang, 19-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝐿 = (𝐶Δfunc𝐷)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    &   (𝜑 → 𝐹 = ( oppFunc ↾ (𝐷 Func 𝐶)))    &   𝑁 = (𝐷 Nat 𝐶)    &   (𝜑 → 𝐺 = (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛𝑁𝑚))))    ⇒   (𝜑 → (⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)) = (𝑂Δfunc𝑃))
 
19-Nov-2025oppfdiag1a 49540 A constant functor for opposite categories is the opposite functor of the constant functor for original categories. (Contributed by Zhi Wang, 19-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝐿 = (𝐶Δfunc𝐷)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    &   𝐴 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐴)    ⇒   (𝜑 → ( oppFunc ‘((1st ‘𝐿)‘𝑋)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋))
 
19-Nov-2025oppfdiag1 49539 A constant functor for opposite categories is the opposite functor of the constant functor for original categories. (Contributed by Zhi Wang, 19-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝐿 = (𝐶Δfunc𝐷)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    &   (𝜑 → 𝐹 = ( oppFunc ↾ (𝐷 Func 𝐶)))    &   𝐴 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐴)    ⇒   (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋))
 
19-Nov-2025fucoppcfunc 49537 A functor from the opposite category of functors to the category of opposite functors. (Contributed by Zhi Wang, 19-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝑄 = (𝐶 FuncCat 𝐷)    &   𝑅 = (oppCat‘𝑄)    &   𝑆 = (𝑂 FuncCat 𝑃)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    ⇒   (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)
 
19-Nov-2025fucoppcffth 49536 A fully faithful functor from the opposite category of functors to the category of opposite functors. (Contributed by Zhi Wang, 19-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝑄 = (𝐶 FuncCat 𝐷)    &   𝑅 = (oppCat‘𝑄)    &   𝑆 = (𝑂 FuncCat 𝑃)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    ⇒   (𝜑 → 𝐹((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆))𝐺)
 
19-Nov-2025opf12 49529 The object part of the op functor on functor categories. Lemma for oppfdiag 49541. (Contributed by Zhi Wang, 19-Nov-2025.)
(𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑁(2nd ‘𝑋)𝑀))
 
19-Nov-2025oppc2ndf 49414 The opposite functor of the second projection functor is the second projection functor of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    ⇒   (𝜑 → ( oppFunc ‘(𝐶 2ndF 𝐷)) = (𝑂 2ndF 𝑃))
 
19-Nov-2025oppc1stf 49413 The opposite functor of the first projection functor is the first projection functor of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    ⇒   (𝜑 → ( oppFunc ‘(𝐶 1stF 𝐷)) = (𝑂 1stF 𝑃))
 
19-Nov-2025oppc1stflem 49412 A utility theorem for proving theorems on projection functors of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    &   ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ( oppFunc ‘(𝐶𝐹𝐷)) = (𝑂𝐹𝑃))    &   𝐹 = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ 𝑌)    ⇒   (𝜑 → ( oppFunc ‘(𝐶𝐹𝐷)) = (𝑂𝐹𝑃))
 
19-Nov-2025uobffth 49343 A fully faithful functor generates equal sets of universal objects. (Contributed by Zhi Wang, 19-Nov-2025.)
𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))    ⇒   (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
 
19-Nov-2025oppf2 49265 Value of the morphism part of the opposite functor. (Contributed by Zhi Wang, 19-Nov-2025.)
(𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑁(2nd ‘𝐹)𝑀))
 
19-Nov-2025oppf1 49264 Value of the object part of the opposite functor. (Contributed by Zhi Wang, 19-Nov-2025.)
(𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → (1st ‘( oppFunc ‘𝐹)) = (1st ‘𝐹))
 
19-Nov-2025oppfval3 49263 Value of the opposite functor. (Contributed by Zhi Wang, 19-Nov-2025.)
(𝜑 → 𝐹 = ⟨𝐺, 𝐾⟩)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → ( oppFunc ‘𝐹) = ⟨𝐺, tpos 𝐾⟩)
 
19-Nov-2025eqfnovd 48990 Deduction for equality of operations. (Contributed by Zhi Wang, 19-Nov-2025.)
(𝜑 → 𝐹 Fn (𝐴 × 𝐵))    &   (𝜑 → 𝐺 Fn (𝐴 × 𝐵))    &   ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))    ⇒   (𝜑 → 𝐹 = 𝐺)
 
19-Nov-2025cos4t3rdpi 42474 The cosine of 4 · (π / 3) is -1 / 2. (Contributed by SN, 19-Nov-2025.)
(cos‘(4 · (π / 3))) = -(1 / 2)
 
19-Nov-2025sin4t3rdpi 42473 The sine of 4 · (π / 3) is -(√3) / 2. (Contributed by SN, 19-Nov-2025.)
(sin‘(4 · (π / 3))) = -((√‘3) / 2)
 
19-Nov-2025cos2t3rdpi 42472 The cosine of 2 · (π / 3) is -1 / 2. (Contributed by SN, 19-Nov-2025.)
(cos‘(2 · (π / 3))) = -(1 / 2)
 
19-Nov-2025sin2t3rdpi 42471 The sine of 2 · (π / 3) is (√3) / 2. (Contributed by SN, 19-Nov-2025.)
(sin‘(2 · (π / 3))) = ((√‘3) / 2)
 
19-Nov-2025cospim 42469 Cosine of a number subtracted from π. (Contributed by SN, 19-Nov-2025.)
(𝐴 ∈ ℂ → (cos‘(π − 𝐴)) = -(cos‘𝐴))
 
19-Nov-2025sinpim 42468 Sine of a number subtracted from π. (Contributed by SN, 19-Nov-2025.)
(𝐴 ∈ ℂ → (sin‘(π − 𝐴)) = (sin‘𝐴))
 
19-Nov-20253rdpwhole 42410 A third of a number plus the number is four thirds of the number. (Contributed by SN, 19-Nov-2025.)
(𝐴 ∈ ℂ → ((𝐴 / 3) + 𝐴) = (4 · (𝐴 / 3)))
 
19-Nov-20251p3e4 42377 1 + 3 = 4. (Contributed by SN, 19-Nov-2025.)
(1 + 3) = 4
 
19-Nov-2025spsv 1988 Generalization of antecedent. A trivial weak version of sps 2190 avoiding ax-12 2182. (Contributed by SN, 13-Nov-2025.) (Proof shortened by WL, 19-Nov-2025.)
(𝜑 → 𝜓)    ⇒   (∀𝑥𝜑 → 𝜓)
 
18-Nov-2025fucoppccic 49538 The opposite category of functors is isomorphic to the category of opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑋 = (oppCat‘(𝐷 FuncCat 𝐸))    &   𝑌 = ((oppCat‘𝐷) FuncCat (oppCat‘𝐸))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐸 ∈ 𝑊)    ⇒   (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌)
 
18-Nov-2025fucoppc 49535 The isomorphism from the opposite category of functors to the category of opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝑄 = (𝐶 FuncCat 𝐷)    &   𝑅 = (oppCat‘𝑄)    &   𝑆 = (𝑂 FuncCat 𝑃)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   𝑇 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝑇)    &   𝐼 = (Iso‘𝑇)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    &   (𝜑 → 𝑅 ∈ 𝐵)    &   (𝜑 → 𝑆 ∈ 𝐵)    ⇒   (𝜑 → 𝐹(𝑅𝐼𝑆)𝐺)
 
18-Nov-2025fucoppcco 49534 The opposite category of functors is compatible with the category of opposite functors in terms of composition. (Contributed by Zhi Wang, 18-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝑄 = (𝐶 FuncCat 𝐷)    &   𝑅 = (oppCat‘𝑄)    &   𝑆 = (𝑂 FuncCat 𝑃)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   (𝜑 → 𝐴 ∈ (𝑋(Hom ‘𝑅)𝑌))    &   (𝜑 → 𝐵 ∈ (𝑌(Hom ‘𝑅)𝑍))    ⇒   (𝜑 → ((𝑋𝐺𝑍)‘(𝐵(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐴)) = (((𝑌𝐺𝑍)‘𝐵)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐴)))
 
18-Nov-2025fucoppcid 49533 The opposite category of functors is compatible with the category of opposite functors in terms of identity morphism. (Contributed by Zhi Wang, 18-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝑄 = (𝐶 FuncCat 𝐷)    &   𝑅 = (oppCat‘𝑄)    &   𝑆 = (𝑂 FuncCat 𝑃)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋)))
 
18-Nov-2025fucoppclem 49532 Lemma for fucoppc 49535. (Contributed by Zhi Wang, 18-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝑌 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → (𝑌𝑁𝑋) = ((𝐹‘𝑋)(𝑂 Nat 𝑃)(𝐹‘𝑌)))
 
18-Nov-2025opf2 49531 The morphism part of the op functor on functor categories. Lemma for fucoppc 49535. (Contributed by Zhi Wang, 18-Nov-2025.)
(𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑦𝑁𝑥))))    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    &   (𝜑 → 𝐷 ∈ (𝑌𝑁𝑋))    ⇒   (𝜑 → ((𝑋𝐹𝑌)‘𝐶) = 𝐷)
 
18-Nov-2025opf2fval 49530 The morphism part of the op functor on functor categories. Lemma for fucoppc 49535. (Contributed by Zhi Wang, 18-Nov-2025.)
(𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑦𝑁𝑥))))    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋𝐹𝑌) = ( I ↾ (𝑌𝑁𝑋)))
 
18-Nov-2025opf11 49528 The object part of the op functor on functor categories. Lemma for fucoppc 49535. (Contributed by Zhi Wang, 18-Nov-2025.)
(𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → (1st ‘(𝐹‘𝑋)) = (1st ‘𝑋))
 
18-Nov-2025natoppfb 49356 A natural transformation is natural between opposite functors, and vice versa. (Contributed by Zhi Wang, 18-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &   𝑀 = (𝑂 Nat 𝑃)    &   (𝜑 → 𝐾 = ( oppFunc ‘𝐹))    &   (𝜑 → 𝐿 = ( oppFunc ‘𝐺))    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    ⇒   (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾))
 
18-Nov-2025natoppf2 49355 A natural transformation is natural between opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &   𝑀 = (𝑂 Nat 𝑃)    &   (𝜑 → 𝐾 = ( oppFunc ‘𝐹))    &   (𝜑 → 𝐿 = ( oppFunc ‘𝐺))    &   (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))    ⇒   (𝜑 → 𝐴 ∈ (𝐿𝑀𝐾))
 
18-Nov-2025natoppf 49354 A natural transformation is natural between opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &   𝑀 = (𝑂 Nat 𝑃)    &   (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))    ⇒   (𝜑 → 𝐴 ∈ (⟨𝐾, tpos 𝐿⟩𝑀⟨𝐹, tpos 𝐺⟩))
 
18-Nov-2025eloppf2 49259 Both components of a pre-image of a non-empty opposite functor exist; and the second component is a relation on triples. (Contributed by Zhi Wang, 18-Nov-2025.)
(𝐹 oppFunc 𝐺) = 𝐾    &   (𝜑 → 𝑋 ∈ 𝐾)    ⇒   (𝜑 → ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Rel 𝐺 ∧ Rel dom 𝐺)))
 
18-Nov-2025eloppf 49258 The pre-image of a non-empty opposite functor is non-empty; and the second component of the pre-image is a relation on triples. (Contributed by Zhi Wang, 18-Nov-2025.)
𝐺 = ( oppFunc ‘𝐹)    &   (𝜑 → 𝑋 ∈ 𝐺)    ⇒   (𝜑 → (𝐹 ≠ ∅ ∧ (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹))))
 
18-Nov-2025pgnbgreunbgrlem3 48242 Lemma 3 for pgnbgreunbgr 48249. (Contributed by AV, 18-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))
 
18-Nov-2025pgnbgreunbgrlem2 48241 Lemma 2 for pgnbgreunbgr 48249. Impossible cases. (Contributed by AV, 18-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ((𝐿 = ⟨1, (((2nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝐿 = ⟨1, (((2nd ‘𝑋) − 2) mod 5)⟩) → ((𝐾 = ⟨1, (((2nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝐾 = ⟨1, (((2nd ‘𝑋) − 2) mod 5)⟩) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
 
18-Nov-2025fxpsdrg 33151 The fixed points of a group action 𝐴 on a division ring 𝑊 is a sub-division-ring. Since sub-division-rings of fields are subfields (see fldsdrgfld 20715), (𝐶FixPts𝐴) might be called the fixed subfield under 𝐴. (Contributed by Thierry Arnoux, 18-Nov-2025.)
𝐵 = (Base‘𝐺)    &   𝐶 = (Base‘𝑊)    &   𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥))    &   (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    &   ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 RingHom 𝑊))    &   (𝜑 → 𝑊 ∈ DivRing)    ⇒   (𝜑 → (𝐶FixPts𝐴) ∈ (SubDRing‘𝑊))
 
18-Nov-2025fxpsubrg 33150 The fixed points of a group action 𝐴 on a ring 𝑊 is a subgring. (Contributed by Thierry Arnoux, 18-Nov-2025.)
𝐵 = (Base‘𝐺)    &   𝐶 = (Base‘𝑊)    &   𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥))    &   (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    &   ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 RingHom 𝑊))    ⇒   (𝜑 → (𝐶FixPts𝐴) ∈ (SubRing‘𝑊))
 
18-Nov-2025fxpsubg 33149 The fixed points of a group action 𝐴 on a group 𝑊 is a subgroup. (Contributed by Thierry Arnoux, 18-Nov-2025.)
𝐵 = (Base‘𝐺)    &   𝐶 = (Base‘𝑊)    &   𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥))    &   (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    &   ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 GrpHom 𝑊))    ⇒   (𝜑 → (𝐶FixPts𝐴) ∈ (SubGrp‘𝑊))
 
18-Nov-2025fxpsubm 33148 Provided the group action 𝐴 induces monoid automorphisms, the set of fixed points of 𝐴 on a monoid 𝑊 is a submonoid, which could be called the fixed submonoid under 𝐴. (Contributed by Thierry Arnoux, 18-Nov-2025.)
𝐵 = (Base‘𝐺)    &   𝐶 = (Base‘𝑊)    &   𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥))    &   (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    &   ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 MndHom 𝑊))    ⇒   (𝜑 → (𝐶FixPts𝐴) ∈ (SubMnd‘𝑊))
 
18-Nov-2025cntrval2 33147 Express the center 𝑍 of a group 𝑀 as the set of fixed points of the conjugation operation ⊕. (Contributed by Thierry Arnoux, 18-Nov-2025.)
𝐵 = (Base‘𝑀)    &    + = (+g‘𝑀)    &    − = (-g‘𝑀)    &    ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥))    &   𝑍 = (Cntr‘𝑀)    ⇒   (𝑀 ∈ Grp → 𝑍 = (𝐵FixPts ⊕ ))
 
18-Nov-2025conjga 33146 Group conjugation induces a group action. (Contributed by Thierry Arnoux, 18-Nov-2025.)
𝐵 = (Base‘𝑀)    &    + = (+g‘𝑀)    &    − = (-g‘𝑀)    &    ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥))    ⇒   (𝑀 ∈ Grp → ⊕ ∈ (𝑀 GrpAct 𝐵))
 
18-Nov-2025fxpgaeq 33145 A fixed point 𝑋 is invariant under group action 𝐴 (Contributed by Thierry Arnoux, 18-Nov-2025.)
𝑈 = (Base‘𝐺)    &   (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    &   (𝜑 → 𝑋 ∈ (𝐶FixPts𝐴))    &   (𝜑 → 𝑃 ∈ 𝑈)    ⇒   (𝜑 → (𝑃𝐴𝑋) = 𝑋)
 
18-Nov-2025isfxp 33144 Property of being a fixed point. (Contributed by Thierry Arnoux, 18-Nov-2025.)
𝑈 = (Base‘𝐺)    &   (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    &   (𝜑 → 𝑋 ∈ 𝐶)    ⇒   (𝜑 → (𝑋 ∈ (𝐶FixPts𝐴) ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
 
18-Nov-2025fxpgaval 33143 Value of the set of fixed points for a group action 𝐴 (Contributed by Thierry Arnoux, 18-Nov-2025.)
𝑈 = (Base‘𝐺)    &   (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    ⇒   (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
 
18-Nov-2025fxpss 33142 The set of fixed points is a subset of the set acted upon. (Contributed by Thierry Arnoux, 18-Nov-2025.)
(𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑊)    ⇒   (𝜑 → (𝐵FixPts𝐴) ⊆ 𝐵)
 
18-Nov-2025fxpval 33141 Value of the set of fixed points. (Contributed by Thierry Arnoux, 18-Nov-2025.)
(𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑊)    ⇒   (𝜑 → (𝐵FixPts𝐴) = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
 
18-Nov-2025df-fxp 33140 Define the set of fixed points left unchanged by a group action. (Contributed by Thierry Arnoux, 18-Nov-2025.)
FixPts = (𝑏 ∈ V, 𝑎 ∈ V ↦ {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥})
 
18-Nov-2025ralimd6v 3186 Deduction sextupally quantifying both antecedent and consequent. (Contributed by Scott Fenton, 5-Mar-2025.) Reduce DV conditions. (Revised by Eric Schmidt, 18-Nov-2025.)
(𝜑 → (𝜓 → 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜒))
 
18-Nov-2025ralimd4v 3184 Deduction quadrupally quantifying both antecedent and consequent. (Contributed by Scott Fenton, 2-Mar-2025.) Reduce DV conditions. (Revised by Eric Schmidt, 18-Nov-2025.)
(𝜑 → (𝜓 → 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜒))
 
18-Nov-2025ralimdvv 3182 Deduction doubly quantifying both antecedent and consequent. (Contributed by Scott Fenton, 2-Mar-2025.) Shorten and reduce DV conditions. (Revised by Eric Schmidt, 18-Nov-2025.)
(𝜑 → (𝜓 → 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))
 
17-Nov-2025initocmd 49794 Initial objects are the object part of colimits of the empty diagram. (Contributed by Zhi Wang, 17-Nov-2025.)
(InitO‘𝐶) = dom (∅(𝐶 Colimit ∅)∅)
 
17-Nov-2025isinito4a 49673 The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 17-Nov-2025.)
(𝜑 → 1 ∈ TermCat)    &   (𝜑 → 𝑋 ∈ (Base‘ 1 ))    &   𝐹 = ((1st ‘( 1 Δfunc𝐶))‘𝑋)    ⇒   (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ dom (𝐹(𝐶 UP 1 )𝑋)))
 
17-Nov-2025isinito4 49672 The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 17-Nov-2025.)
(𝜑 → 1 ∈ TermCat)    &   (𝜑 → 𝑋 ∈ (Base‘ 1 ))    &   (𝜑 → 𝐹 ∈ (𝐶 Func 1 ))    ⇒   (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ dom (𝐹(𝐶 UP 1 )𝑋)))
 
17-Nov-2025uobeqterm 49671 Universal objects and terminal categories. (Contributed by Zhi Wang, 17-Nov-2025.)
𝐴 = (Base‘𝐷)    &   𝐵 = (Base‘𝐸)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    &   (𝜑 → 𝐷 ∈ TermCat)    &   (𝜑 → 𝐸 ∈ TermCat)    ⇒   (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
 
17-Nov-2025cofuterm 49670 Post-compose with a functor to a terminal category. (Contributed by Zhi Wang, 17-Nov-2025.)
(𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐾 ∈ (𝐶 Func 𝐸))    &   (𝜑 → 𝐸 ∈ TermCat)    ⇒   (𝜑 → (𝐺 ∘func 𝐹) = 𝐾)
 
17-Nov-2025termfucterm 49669 All functors between two terminal categories are isomorphisms. (Contributed by Zhi Wang, 17-Nov-2025.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝐼 = (Iso‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑋 ∈ TermCat)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ TermCat)    ⇒   (𝜑 → (𝑋 Func 𝑌) = (𝑋𝐼𝑌))
 
17-Nov-20250fucterm 49668 The category of functors from an initial category is terminal. (Contributed by Zhi Wang, 17-Nov-2025.)
(𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → ∅ = (Base‘𝐶))    &   (𝜑 → 𝐷 ∈ Cat)    &   𝑄 = (𝐶 FuncCat 𝐷)    ⇒   (𝜑 → 𝑄 ∈ TermCat)
 
17-Nov-2025fucterm 49667 The category of functors to a terminal category is terminal. (Contributed by Zhi Wang, 17-Nov-2025.)
𝑄 = (𝐶 FuncCat 𝐷)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ TermCat)    ⇒   (𝜑 → 𝑄 ∈ TermCat)
 
17-Nov-2025funcsn 49666 The category of one functor to a thin category is terminal. (Contributed by Zhi Wang, 17-Nov-2025.)
𝑄 = (𝐶 FuncCat 𝐷)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → (𝐶 Func 𝐷) = {𝐹})    &   (𝜑 → 𝐷 ∈ ThinCat)    ⇒   (𝜑 → 𝑄 ∈ TermCat)
 
17-Nov-2025termco 49606 The object of a terminal category. (Contributed by Zhi Wang, 17-Nov-2025.)
(𝜑 → 𝐶 ∈ TermCat)    &   𝐵 = (Base‘𝐶)    ⇒   (𝜑 → ∪ 𝐵 ∈ 𝐵)
 
17-Nov-2025uobeq3 49527 An isomorphism between categories generates equal sets of universal objects. (Contributed by Zhi Wang, 17-Nov-2025.)
𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   𝑄 = (CatCat‘𝑈)    &   𝐼 = (Iso‘𝑄)    &   (𝜑 → 𝐾 ∈ (𝐷𝐼𝐸))    ⇒   (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
 
17-Nov-2025uobeq2 49526 If a full functor (in fact, a full embedding) is a section, then the sets of universal objects are equal. (Contributed by Zhi Wang, 17-Nov-2025.)
𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   𝑄 = (CatCat‘𝑈)    &   𝑆 = (Sect‘𝑄)    &   (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸))    &   (𝜑 → 𝐾 ∈ dom (𝐷𝑆𝐸))    ⇒   (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
 
17-Nov-2025catcisoi 49525 A functor is an isomorphism of categories only if it is full and faithful, and is a bijection on the objects. Remark 3.28(2) in [Adamek] p. 34. (Contributed by Zhi Wang, 17-Nov-2025.)
𝐶 = (CatCat‘𝑈)    &   𝑅 = (Base‘𝑋)    &   𝑆 = (Base‘𝑌)    &   𝐼 = (Iso‘𝐶)    &   (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    ⇒   (𝜑 → (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆))
 
17-Nov-2025uobeq 49345 If a full functor (in fact, a full embedding) is a section of a functor (surjective on objects), then the sets of universal objects are equal. (Contributed by Zhi Wang, 17-Nov-2025.)
𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   𝐼 = (idfunc‘𝐷)    &   (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸))    &   (𝜑 → (𝐿 ∘func 𝐾) = 𝐼)    &   (𝜑 → 𝐿 ∈ (𝐸 Func 𝐷))    ⇒   (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
 
17-Nov-2025uobeqw 49344 If a full functor (in fact, a full embedding) is a section of a fully faithful functor (surjective on objects), then the sets of universal objects are equal. (Contributed by Zhi Wang, 17-Nov-2025.)
𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   𝐼 = (idfunc‘𝐷)    &   (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸))    &   (𝜑 → (𝐿 ∘func 𝐾) = 𝐼)    &   (𝜑 → 𝐿 ∈ ((𝐸 Full 𝐷) ∩ (𝐸 Faith 𝐷)))    ⇒   (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
 
17-Nov-2025uptrar 49341 Universal property and fully faithful functor. (Contributed by Zhi Wang, 17-Nov-2025.)
(𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))    &   (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑁) = 𝑀)    &   (𝜑 → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)    ⇒   (𝜑 → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)
 
17-Nov-2025uobrcl 49318 Reverse closure for universal object. (Contributed by Zhi Wang, 17-Nov-2025.)
(𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
 
17-Nov-2025oppff1o 49274 The operation generating opposite functors is bijective. (Contributed by Zhi Wang, 17-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    ⇒   (𝜑 → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))
 
17-Nov-2025oppff1 49273 The operation generating opposite functors is injective. (Contributed by Zhi Wang, 17-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    ⇒   ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1→(𝑂 Func 𝑃)
 
17-Nov-20252oppffunc 49271 The opposite functor of an opposite functor is a functor on the original categories. (Contributed by Zhi Wang, 14-Nov-2025.) The functor in opposite categories does not have to be an opposite functor. (Revised by Zhi Wang, 17-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    &   (𝜑 → 𝐹 ∈ (𝑂 Func 𝑃))    ⇒   (𝜑 → ( oppFunc ‘𝐹) ∈ (𝐶 Func 𝐷))
 
17-Nov-2025oppffn 49249 oppFunc is a function on (V × V). (Contributed by Zhi Wang, 17-Nov-2025.)
oppFunc Fn (V × V)
 
17-Nov-2025isoval2 49160 The isomorphisms are the domain of the inverse relation. (Contributed by Zhi Wang, 17-Nov-2025.)
𝑁 = (Inv‘𝐶)    &   𝐼 = (Iso‘𝐶)    ⇒   (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌)
 
17-Nov-2025isorcl2 49159 Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.)
𝐼 = (Iso‘𝐶)    &   (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    &   𝐵 = (Base‘𝐶)    ⇒   (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
 
17-Nov-2025isorcl 49158 Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.)
𝐼 = (Iso‘𝐶)    &   (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    ⇒   (𝜑 → 𝐶 ∈ Cat)
 
17-Nov-2025pgnbgreunbgrlem2lem3 48240 Lemma 3 for pgnbgreunbgrlem2 48241. (Contributed by AV, 17-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ((((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸)
 
16-Nov-2025uptrai 49342 Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.)
(𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))    &   (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)    &   (𝜑 → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)    ⇒   (𝜑 → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)
 
16-Nov-2025uptra 49340 Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.)
(𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))    &   (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑 → 𝑀 ∈ (𝑋𝐽((1st ‘𝐹)‘𝑍)))    ⇒   (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁))
 
16-Nov-2025uptri 49339 Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.)
(𝜑 → (𝑅‘𝑋) = 𝑌)    &   (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)    &   (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)    &   (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)    &   (𝜑 → 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀)    ⇒   (𝜑 → 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁)
 
16-Nov-2025uptr 49338 Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.)
(𝜑 → (𝑅‘𝑋) = 𝑌)    &   (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)    &   (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑 → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍)))    ⇒   (𝜑 → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁))
 
16-Nov-2025uptrlem3 49337 Lemma for uptr 49338. (Contributed by Zhi Wang, 16-Nov-2025.)
(𝜑 → (𝑅‘𝑋) = 𝑌)    &   (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)    &   (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑 → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍)))    &   𝐴 = (Base‘𝐶)    &   (𝜑 → 𝑍 ∈ 𝐴)    ⇒   (𝜑 → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁))
 
16-Nov-2025uptrlem2 49336 Lemma for uptr 49338. (Contributed by Zhi Wang, 16-Nov-2025.)
𝐻 = (Hom ‘𝐶)    &   𝐼 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &    ∙ = (comp‘𝐷)    &    ⚬ = (comp‘𝐸)    &   𝐴 = (Base‘𝐶)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   (𝜑 → 𝑍 ∈ 𝐴)    &   (𝜑 → 𝑊 ∈ 𝐴)    &   (𝜑 → 𝑀 ∈ (𝑋𝐼((1st ‘𝐹)‘𝑍)))    &   (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))    &   (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    ⇒   (𝜑 → (∀ℎ ∈ (𝑌𝐽((1st ‘𝐺)‘𝑊))∃!𝑘 ∈ (𝑍𝐻𝑊)ℎ = (((𝑍(2nd ‘𝐺)𝑊)‘𝑘)(⟨𝑌, ((1st ‘𝐺)‘𝑍)⟩ ⚬ ((1st ‘𝐺)‘𝑊))𝑁) ↔ ∀𝑔 ∈ (𝑋𝐼((1st ‘𝐹)‘𝑊))∃!𝑘 ∈ (𝑍𝐻𝑊)𝑔 = (((𝑍(2nd ‘𝐹)𝑊)‘𝑘)(⟨𝑋, ((1st ‘𝐹)‘𝑍)⟩ ∙ ((1st ‘𝐹)‘𝑊))𝑀)))
 
16-Nov-2025uptrlem1 49335 Lemma for uptr 49338. (Contributed by Zhi Wang, 16-Nov-2025.)
𝐻 = (Hom ‘𝐶)    &   𝐼 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &    ∙ = (comp‘𝐷)    &    ⚬ = (comp‘𝐸)    &   (𝜑 → 𝑋 ∈ (Base‘𝐷))    &   (𝜑 → (𝑀‘𝑋) = 𝑌)    &   (𝜑 → 𝑍 ∈ (Base‘𝐶))    &   (𝜑 → 𝑊 ∈ (Base‘𝐶))    &   (𝜑 → 𝐴 ∈ (𝑋𝐼(𝐹‘𝑍)))    &   (𝜑 → ((𝑋𝑁(𝐹‘𝑍))‘𝐴) = 𝐵)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   (𝜑 → 𝑀((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑁)    &   (𝜑 → (⟨𝑀, 𝑁⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)    ⇒   (𝜑 → (∀ℎ ∈ (𝑌𝐽(𝐾‘𝑊))∃!𝑘 ∈ (𝑍𝐻𝑊)ℎ = (((𝑍𝐿𝑊)‘𝑘)(⟨𝑌, (𝐾‘𝑍)⟩ ⚬ (𝐾‘𝑊))𝐵) ↔ ∀𝑔 ∈ (𝑋𝐼(𝐹‘𝑊))∃!𝑘 ∈ (𝑍𝐻𝑊)𝑔 = (((𝑍𝐺𝑊)‘𝑘)(⟨𝑋, (𝐹‘𝑍)⟩ ∙ (𝐹‘𝑊))𝐴)))
 
16-Nov-2025idemb 49284 The inclusion functor is an embedding. Remark 4.4(1) in [Adamek] p. 49. (Contributed by Zhi Wang, 16-Nov-2025.)
𝐼 = (idfunc‘𝐶)    ⇒   (𝐼 ∈ (𝐷 Func 𝐸) → (𝐼 ∈ (𝐷 Faith 𝐸) ∧ Fun ◡(1st ‘𝐼)))
 
16-Nov-2025idfu1stf1o 49224 The identity functor/inclusion functor is bijective on objects. (Contributed by Zhi Wang, 16-Nov-2025.)
𝐼 = (idfunc‘𝐶)    &   𝐵 = (Base‘𝐶)    ⇒   (𝐶 ∈ Cat → (1st ‘𝐼):𝐵–1-1-onto→𝐵)
 
16-Nov-2025cofucla 49221 The composition of two functors is a functor. Proposition 3.23 of [Adamek] p. 33. (Contributed by Zhi Wang, 16-Nov-2025.)
(𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    ⇒   (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) ∈ (𝐶 Func 𝐸))
 
16-Nov-2025cofu2a 49220 Value of the morphism part of the functor composition. (Contributed by Zhi Wang, 16-Nov-2025.)
𝐵 = (Base‘𝐶)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝑀, 𝑁⟩)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    ⇒   (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑋𝑁𝑌)‘𝑅))
 
16-Nov-2025cofu1a 49219 Value of the object part of the functor composition. (Contributed by Zhi Wang, 16-Nov-2025.)
𝐵 = (Base‘𝐶)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝑀, 𝑁⟩)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝐾‘(𝐹‘𝑋)) = (𝑀‘𝑋))
 
16-Nov-2025pgnbgreunbgrlem2lem2 48239 Lemma 2 for pgnbgreunbgrlem2 48241. (Contributed by AV, 16-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ((((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸)
 
16-Nov-2025pgnbgreunbgrlem2lem1 48238 Lemma 1 for pgnbgreunbgrlem2 48241. (Contributed by AV, 16-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ((((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸)
 
16-Nov-2025nregmodelaxext 45135 The Axiom of Extensionality ax-ext 2705 is true in the permutation model defined from 𝐹. This theorem is an immediate consequence of the fact that ax-ext 2705 holds in all permutation models and is provided as an illustration. (Contributed by Eric Schmidt, 16-Nov-2025.)
𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})    &   𝑅 = (◡𝐹 ∘ E )    ⇒   (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)
 
16-Nov-2025nregmodel 45134 The Axiom of Regularity ax-reg 9485 is false in the permutation model defined from 𝐹. Since the other axioms of ZFC hold in all permutation models (permaxext 45122 through permac8prim 45131), we can conclude that Regularity does not follow from those axioms, assuming ZFC is consistent. (If we could prove Regularity from the other axioms, we could prove it in the permutation model and thus obtain a contradiction with this theorem.) Since we also know that Regularity is consistent with the other axioms (wfaxext 45110 through wfac8prim 45119), Regularity is neither provable nor disprovable from the other axioms; i.e., it is independent of them. (Contributed by Eric Schmidt, 16-Nov-2025.)
𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})    &   𝑅 = (◡𝐹 ∘ E )    ⇒    ¬ ∀𝑥(∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥)))
 
16-Nov-2025nregmodellem 45133 Lemma for nregmodel 45134. (Contributed by Eric Schmidt, 16-Nov-2025.)
𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})    &   𝑅 = (◡𝐹 ∘ E )    ⇒   (𝑥𝑅∅ ↔ 𝑥 ∈ {∅})
 
16-Nov-2025nregmodelf1o 45132 Define a permutation 𝐹 used to produce a model in which ax-reg 9485 is false. The permutation swaps ∅ and {∅} and leaves the rest of 𝑉 fixed. This is an example given after Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 16-Nov-2025.)
𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})    ⇒   𝐹:V–1-1-onto→V
 
16-Nov-2025permac8prim 45131 The Axiom of Choice ac8prim 45108 holds in permutation models. Part of Exercise II.9.3 of [Kunen2] p. 149. Note that ax-ac 10357 requires Regularity for its derivation from the usual Axiom of Choice and does not necessarily hold in permutation models. (Contributed by Eric Schmidt, 16-Nov-2025.)
𝐹:V–1-1-onto→V    &   𝑅 = (◡𝐹 ∘ E )    ⇒   ((∀𝑧(𝑧𝑅𝑥 → ∃𝑤 𝑤𝑅𝑧) ∧ ∀𝑧∀𝑤((𝑧𝑅𝑥 ∧ 𝑤𝑅𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤)))) → ∃𝑦∀𝑧(𝑧𝑅𝑥 → ∃𝑤∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤)))
 
15-Nov-2025cofidfth 49287 If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then 𝐹 is faithful. Combined with cofidf1 49246, this theorem proves that 𝐹 is an embedding (a faithful functor injective on objects, remark 3.28(1) of [Adamek] p. 34). (Contributed by Zhi Wang, 15-Nov-2025.)
𝐼 = (idfunc‘𝐷)    &   (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)    &   (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)    ⇒   (𝜑 → 𝐹(𝐷 Faith 𝐸)𝐺)
 
15-Nov-2025cofidf1 49246 If "⟨𝐹, 𝐺⟩ is a section of ⟨𝐾, 𝐿⟩ " in a category of small categories (in a universe), then 𝐹 is injective, and 𝐾 is surjective. (Contributed by Zhi Wang, 15-Nov-2025.)
𝐼 = (idfunc‘𝐷)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)    &   (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)    &   𝐶 = (Base‘𝐸)    ⇒   (𝜑 → (𝐹:𝐵–1-1→𝐶 ∧ 𝐾:𝐶–onto→𝐵))
 
15-Nov-2025cofidf2 49245 If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then the morphism part of 𝐹 is injective, and the morphism part of 𝐺 is surjective in the image of 𝐹. (Contributed by Zhi Wang, 15-Nov-2025.)
𝐼 = (idfunc‘𝐷)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)    &   (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∧ ((𝐹‘𝑋)𝐿(𝐹‘𝑌)):((𝐹‘𝑋)𝐽(𝐹‘𝑌))–onto→(𝑋𝐻𝑌)))
 
15-Nov-2025cofidval 49244 The property "⟨𝐹, 𝐺⟩ is a section of ⟨𝐾, 𝐿⟩ " in a category of small categories (in a universe); expressed explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)
𝐼 = (idfunc‘𝐷)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)    &   (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)    &   𝐻 = (Hom ‘𝐷)    ⇒   (𝜑 → ((𝐾 ∘ 𝐹) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))))
 
15-Nov-2025cofidf1a 49243 If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then the object part of 𝐹 is injective, and the object part of 𝐺 is surjective. (Contributed by Zhi Wang, 15-Nov-2025.)
𝐼 = (idfunc‘𝐷)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))    &   (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)    &   𝐶 = (Base‘𝐸)    ⇒   (𝜑 → ((1st ‘𝐹):𝐵–1-1→𝐶 ∧ (1st ‘𝐺):𝐶–onto→𝐵))
 
15-Nov-2025cofidf2a 49242 If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then the morphism part of 𝐹 is injective, and the morphism part of 𝐺 is surjective in the image of 𝐹. (Contributed by Zhi Wang, 15-Nov-2025.)
𝐼 = (idfunc‘𝐷)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))    &   (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋(2nd ‘𝐹)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘𝐹)‘𝑋)𝐽((1st ‘𝐹)‘𝑌)) ∧ (((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)):(((1st ‘𝐹)‘𝑋)𝐽((1st ‘𝐹)‘𝑌))–onto→(𝑋𝐻𝑌)))
 
15-Nov-2025cofidvala 49241 The property "𝐹 is a section of 𝐺 " in a category of small categories (in a universe); expressed explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)
𝐼 = (idfunc‘𝐷)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))    &   (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)    &   𝐻 = (Hom ‘𝐷)    ⇒   (𝜑 → (((1st ‘𝐺) ∘ (1st ‘𝐹)) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))))
 
15-Nov-2025cofid2 49240 Express the morphism part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)
𝐼 = (idfunc‘𝐷)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)    &   (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐻 = (Hom ‘𝐷)    &   (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    ⇒   (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = 𝑅)
 
15-Nov-2025cofid1 49239 Express the object part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)
𝐼 = (idfunc‘𝐷)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)    &   (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)    ⇒   (𝜑 → (𝐾‘(𝐹‘𝑋)) = 𝑋)
 
15-Nov-2025cofid2a 49238 Express the morphism part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)
𝐼 = (idfunc‘𝐷)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))    &   (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐻 = (Hom ‘𝐷)    &   (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    ⇒   (𝜑 → ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)) = 𝑅)
 
15-Nov-2025cofid1a 49237 Express the object part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)
𝐼 = (idfunc‘𝐷)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))    &   (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)    ⇒   (𝜑 → ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)) = 𝑋)
 
15-Nov-2025cofu1st2nd 49217 Rewrite the functor composition with separated functor parts. (Contributed by Zhi Wang, 15-Nov-2025.)
(𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    ⇒   (𝜑 → (𝐺 ∘func 𝐹) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
 
15-Nov-2025initc 49216 Sets with empty base are the only initial objects in the category of small categories. Example 7.2(3) of [Adamek] p. 101. (Contributed by Zhi Wang, 15-Nov-2025.)
((𝐶 ∈ V ∧ ∅ = (Base‘𝐶)) ↔ ∀𝑑 ∈ Cat ∃!𝑓 𝑓 ∈ (𝐶 Func 𝑑))
 
15-Nov-2025func2nd 49203 Extract the second member of a functor. (Contributed by Zhi Wang, 15-Nov-2025.)
(𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
 
15-Nov-2025func1st 49202 Extract the first member of a functor. (Contributed by Zhi Wang, 15-Nov-2025.)
(𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
 
15-Nov-2025pgnbgreunbgrlem1 48237 Lemma 1 for pgnbgreunbgr 48249. (Contributed by AV, 15-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ((𝐿 = ⟨0, (((2nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2nd ‘𝑋) − 1) mod 5)⟩) → ((𝐾 = ⟨0, (((2nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2nd ‘𝑋) − 1) mod 5)⟩) → ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
 
15-Nov-2025gpgedg2ov 48190 The edges of the generalized Petersen graph GPG(N,K) between two outside vertices. (Contributed by AV, 15-Nov-2025.)
𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   𝐼 = (0..^𝑁)    &   𝐺 = (𝑁 gPetersenGr 𝐾)    &   𝐸 = (Edg‘𝐺)    ⇒   (((𝑁 ∈ (ℤ≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸) ↔ 𝑋 = 𝑌))
 
15-Nov-2025modm1p1ne 47494 If an integer minus one equals another integer plus one modulo an integer greater than 4, then the first integer plus one is not equal to the second integer minus one modulo the same modulus. (Contributed by AV, 15-Nov-2025.)
𝐼 = (0..^𝑁)    ⇒   ((𝑁 ∈ (ℤ≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) → ((𝑌 + 1) mod 𝑁) ≠ ((𝑋 − 1) mod 𝑁)))
 
15-Nov-2025modm1nep1 47489 A nonnegative integer less than a modulus greater than 2 plus/minus one are not equal modulo the modulus. (Contributed by AV, 15-Nov-2025.)
𝐼 = (0..^𝑁)    ⇒   ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑌 ∈ 𝐼) → ((𝑌 − 1) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁))
 
15-Nov-2025mod2addne 47488 The sums of a nonnegative integer less than the modulus and two integers whose difference is less than the modulus are not equal modulo the modulus. (Contributed by AV, 15-Nov-2025.)
𝐼 = (0..^𝑁)    ⇒   ((𝑁 ∈ ℕ ∧ (𝑋 ∈ 𝐼 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (abs‘(𝐴 − 𝐵)) ∈ (1..^𝑁)) → ((𝑋 + 𝐴) mod 𝑁) ≠ ((𝑋 + 𝐵) mod 𝑁))
 
15-Nov-2025modmknepk 47486 A nonnegative integer less than the modulus plus/minus a positive integer less than (the ceiling of) half of the modulus are not equal modulo the modulus. For this theorem, it is essential that 𝐾 < (𝑁 / 2)! (Contributed by AV, 3-Sep-2025.) (Revised by AV, 15-Nov-2025.)
𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   𝐼 = (0..^𝑁)    ⇒   ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑌 ∈ 𝐼 ∧ 𝐾 ∈ 𝐽) → ((𝑌 − 𝐾) mod 𝑁) ≠ ((𝑌 + 𝐾) mod 𝑁))
 
15-Nov-2025modmkpkne 47485 If an integer minus a constant equals another integer plus the constant modulo 𝑁, then the first integer plus the constant equals the second integer minus the constant modulo 𝑁 iff the fourfold of the constant is a multiple of 𝑁. (Contributed by AV, 15-Nov-2025.)
((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 − 𝐾) mod 𝑁) = ((𝑋 + 𝐾) mod 𝑁) → (((𝑌 + 𝐾) mod 𝑁) = ((𝑋 − 𝐾) mod 𝑁) ↔ ((4 · 𝐾) mod 𝑁) = 0)))
 
15-Nov-2025trisecnconstr 33826 Not all angles can be trisected. (Contributed by Thierry Arnoux, 15-Nov-2025.)
¬ ∀𝑜 ∈ Constr (𝑜↑𝑐(1 / 3)) ∈ Constr
 
15-Nov-2025cos9thpinconstr 33825 Trisecting an angle is an impossible construction. Given for example 𝑂 = (exp‘((i · (2 · π)) / 3)), which represents an angle of ((2 · π) / 3), the cube root of 𝑂 is not constructible with straightedge and compass, while 𝑂 itself is constructible. This is the second part of Metamath 100 proof #8. Theorem 7.14 of [Stewart] p. 99. (Contributed by Thierry Arnoux and Saveliy Skresanov, 15-Nov-2025.)
𝑂 = (exp‘((i · (2 · π)) / 3))    &   𝑍 = (𝑂↑𝑐(1 / 3))    ⇒   (𝑂 ∈ Constr ∧ 𝑍 ∉ Constr)
 
15-Nov-2025cos9thpinconstrlem2 33824 The complex number 𝐴 is not constructible. (Contributed by Thierry Arnoux, 15-Nov-2025.)
𝑂 = (exp‘((i · (2 · π)) / 3))    &   𝑍 = (𝑂↑𝑐(1 / 3))    &   𝐴 = (𝑍 + (1 / 𝑍))    ⇒    ¬ 𝐴 ∈ Constr
 
15-Nov-2025difmod0 16200 The difference of two integers modulo a positive integer equals zero iff the two integers are equal modulo the positive integer. (Contributed by AV, 15-Nov-2025.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 − 𝐵) mod 𝑁) = 0 ↔ (𝐴 mod 𝑁) = (𝐵 mod 𝑁)))
 
15-Nov-2025uzuzle35 12787 An integer greater than or equal to 5 is an integer greater than or equal to 3. (Contributed by AV, 15-Nov-2025.)
(𝐴 ∈ (ℤ≥‘5) → 𝐴 ∈ (ℤ≥‘3))
 
15-Nov-2025addsubsub23 11532 Swap the second and the third terms in a difference of a sum and a difference (or, vice versa, in a sum of a difference and a sum). (Contributed by AV, 15-Nov-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐷 ∈ ℂ)    ⇒   (𝜑 → ((𝐴 + 𝐵) − (𝐶 − 𝐷)) = ((𝐴 − 𝐶) + (𝐵 + 𝐷)))
 
15-Nov-2025subsubadd23 11531 Swap the second and the third terms in a difference of a difference and a sum. (Contributed by AV, 15-Nov-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐷 ∈ ℂ)    ⇒   (𝜑 → ((𝐴 − 𝐵) − (𝐶 + 𝐷)) = ((𝐴 − 𝐶) − (𝐵 + 𝐷)))
 
14-Nov-2025islmd 49790 The universal property of limits of a diagram. (Contributed by Zhi Wang, 14-Nov-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   𝐴 = (Base‘𝐶)    &   𝑁 = (𝐷 Nat 𝐶)    &   𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    ⇒   (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑅 ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹)∃!𝑚 ∈ (𝑥𝐻𝑋)𝑎 = (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗))𝑚))))
 
14-Nov-2025rellmd 49784 The set of limits of a diagram is a relation. (Contributed by Zhi Wang, 14-Nov-2025.)
Rel ((𝐶 Limit 𝐷)‘𝐹)
 
14-Nov-2025lmdfval2 49780 The set of limits of a diagram. (Contributed by Zhi Wang, 14-Nov-2025.)
((𝐶 Limit 𝐷)‘𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
 
14-Nov-2025reldmlmd2 49778 The domain of (𝐶 Limit 𝐷) is a relation. (Contributed by Zhi Wang, 14-Nov-2025.)
Rel dom (𝐶 Limit 𝐷)
 
14-Nov-2025lmdfval 49774 Function value of Limit. (Contributed by Zhi Wang, 14-Nov-2025.)
(𝐶 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓))
 
14-Nov-2025catcinv 49524 The property "𝐹 is an inverse of 𝐺 " in a category of small categories (in a universe). (Contributed by Zhi Wang, 14-Nov-2025.)
𝐶 = (CatCat‘𝑈)    &   𝑁 = (Inv‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (idfunc‘𝑋)    &   𝐽 = (idfunc‘𝑌)    ⇒   (𝐹(𝑋𝑁𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ ((𝐺 ∘func 𝐹) = 𝐼 ∧ (𝐹 ∘func 𝐺) = 𝐽)))
 
14-Nov-2025catcsect 49523 The property "𝐹 is a section of 𝐺 " in a category of small categories (in a universe). (Contributed by Zhi Wang, 14-Nov-2025.)
𝐶 = (CatCat‘𝑈)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (idfunc‘𝑋)    &   𝑆 = (Sect‘𝐶)    ⇒   (𝐹(𝑋𝑆𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺 ∘func 𝐹) = 𝐼))
 
14-Nov-2025elcatchom 49522 A morphism of the category of categories (in a universe) is a functor. See df-catc 18008 for the definition of the category Cat, which consists of all categories in the universe 𝑢 (i.e., "𝑢-small categories", see Definition 3.44. of [Adamek] p. 39), with functors as the morphisms (catchom 18012). (Contributed by Zhi Wang, 14-Nov-2025.)
𝐶 = (CatCat‘𝑈)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   (𝜑 → 𝐹 ∈ (𝑋 Func 𝑌))
 
14-Nov-2025catcrcl2 49521 Reverse closure for the category of categories (in a universe) (Contributed by Zhi Wang, 14-Nov-2025.)
𝐶 = (CatCat‘𝑈)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   𝐵 = (Base‘𝐶)    ⇒   (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
 
14-Nov-2025catcrcl 49520 Reverse closure for the category of categories (in a universe) (Contributed by Zhi Wang, 14-Nov-2025.)
𝐶 = (CatCat‘𝑈)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   (𝜑 → 𝑈 ∈ V)
 
14-Nov-2025oppfuprcl2 49330 Reverse closure for the class of universal property for opposite functors. (Contributed by Zhi Wang, 14-Nov-2025.)
(𝜑 → 𝑋(𝐺(𝑂 UP 𝑃)𝑊)𝑀)    &   𝐺 = ( oppFunc ‘𝐹)    &   𝑂 = (oppCat‘𝐷)    &   𝑃 = (oppCat‘𝐸)    &   (𝜑 → 𝐷 ∈ 𝑈)    &   (𝜑 → 𝐸 ∈ 𝑉)    &   (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)    ⇒   (𝜑 → 𝐴(𝐷 Func 𝐸)𝐵)
 
14-Nov-2025oppfuprcl 49329 Reverse closure for the class of universal property for opposite functors. (Contributed by Zhi Wang, 14-Nov-2025.)
(𝜑 → 𝑋(𝐺(𝑂 UP 𝑃)𝑊)𝑀)    &   𝐺 = ( oppFunc ‘𝐹)    &   𝑂 = (oppCat‘𝐷)    &   𝑃 = (oppCat‘𝐸)    &   (𝜑 → 𝐷 ∈ 𝑈)    &   (𝜑 → 𝐸 ∈ 𝑉)    ⇒   (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
 
14-Nov-2025uprcl2a 49328 Reverse closure for the class of universal property. (Contributed by Zhi Wang, 14-Nov-2025.)
(𝜑 → 𝑋(𝐺(𝑂 UP 𝑃)𝑊)𝑀)    ⇒   (𝜑 → 𝐺 ∈ (𝑂 Func 𝑃))
 
14-Nov-2025funcoppc5 49270 A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 14-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    &   (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))    ⇒   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
 
14-Nov-2025funcoppc4 49269 A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 14-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    &   (𝜑 → (𝐹 oppFunc 𝐺) ∈ (𝑂 Func 𝑃))    ⇒   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
 
14-Nov-2025oppfoppc2 49267 The opposite functor is a functor on opposite categories. (Contributed by Zhi Wang, 14-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))
 
14-Nov-20252oppf 49257 The double opposite functor is the original functor. Remark 3.42 of [Adamek] p. 39. (Contributed by Zhi Wang, 14-Nov-2025.)
(𝜑 → 𝐺 ∈ 𝑅)    &   Rel 𝑅    &   𝐺 = ( oppFunc ‘𝐹)    ⇒   (𝜑 → ( oppFunc ‘𝐺) = 𝐹)
 
14-Nov-2025oppf1st2nd 49256 Rewrite the opposite functor into its components (eqopi 7963). (Contributed by Zhi Wang, 14-Nov-2025.)
(𝜑 → 𝐺 ∈ 𝑅)    &   Rel 𝑅    &   𝐺 = ( oppFunc ‘𝐹)    &   (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)    ⇒   (𝜑 → (𝐺 ∈ (V × V) ∧ ((1st ‘𝐺) = 𝐴 ∧ (2nd ‘𝐺) = tpos 𝐵)))
 
14-Nov-2025oppfrcl3 49255 If an opposite functor of a class is a functor, then the second component of the original class must be a relation whose domain is a relation as well. (Contributed by Zhi Wang, 14-Nov-2025.)
(𝜑 → 𝐺 ∈ 𝑅)    &   Rel 𝑅    &   𝐺 = ( oppFunc ‘𝐹)    &   (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)    ⇒   (𝜑 → (Rel 𝐵 ∧ Rel dom 𝐵))
 
14-Nov-2025oppfrcl2 49254 If an opposite functor of a class is a functor, then the two components of the original class must be sets. (Contributed by Zhi Wang, 14-Nov-2025.)
(𝜑 → 𝐺 ∈ 𝑅)    &   Rel 𝑅    &   𝐺 = ( oppFunc ‘𝐹)    &   (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)    ⇒   (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
14-Nov-2025oppfrcl 49253 If an opposite functor of a class is a functor, then the original class must be an ordered pair. (Contributed by Zhi Wang, 14-Nov-2025.)
(𝜑 → 𝐺 ∈ 𝑅)    &   Rel 𝑅    &   𝐺 = ( oppFunc ‘𝐹)    ⇒   (𝜑 → 𝐹 ∈ (V × V))
 
14-Nov-2025oppfrcllem 49252 Lemma for oppfrcl 49253. (Contributed by Zhi Wang, 14-Nov-2025.)
(𝜑 → 𝐺 ∈ 𝑅)    &   Rel 𝑅    ⇒   (𝜑 → 𝐺 ≠ ∅)
 
14-Nov-2025isinv2 49151 The property "𝐹 is an inverse of 𝐺". (Contributed by Zhi Wang, 14-Nov-2025.)
𝑁 = (Inv‘𝐶)    &   𝑆 = (Sect‘𝐶)    ⇒   (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹))
 
14-Nov-2025invrcl2 49150 Reverse closure for inverse relations. (Contributed by Zhi Wang, 14-Nov-2025.)
𝑁 = (Inv‘𝐶)    &   (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)    &   𝐵 = (Base‘𝐶)    ⇒   (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
 
14-Nov-2025invrcl 49149 Reverse closure for inverse relations. (Contributed by Zhi Wang, 14-Nov-2025.)
𝑁 = (Inv‘𝐶)    &   (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)    ⇒   (𝜑 → 𝐶 ∈ Cat)
 
14-Nov-2025sectrcl2 49148 Reverse closure for section relations. (Contributed by Zhi Wang, 14-Nov-2025.)
𝑆 = (Sect‘𝐶)    &   (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)    &   𝐵 = (Base‘𝐶)    ⇒   (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
 
14-Nov-2025sectrcl 49147 Reverse closure for section relations. (Contributed by Zhi Wang, 14-Nov-2025.)
𝑆 = (Sect‘𝐶)    &   (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)    ⇒   (𝜑 → 𝐶 ∈ Cat)
 
14-Nov-2025cos9thpinconstrlem1 33823 The complex number 𝑂, representing an angle of (2 · π) / 3, is constructible. (Contributed by Thierry Arnoux, 14-Nov-2025.)
𝑂 = (exp‘((i · (2 · π)) / 3))    ⇒   𝑂 ∈ Constr
 
14-Nov-2025cos9thpiminply 33822 The polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)) is the minimal polynomial for 𝐴 over ℚ, and its degree is 3. (Contributed by Thierry Arnoux, 14-Nov-2025.)
𝑂 = (exp‘((i · (2 · π)) / 3))    &   𝑍 = (𝑂↑𝑐(1 / 3))    &   𝐴 = (𝑍 + (1 / 𝑍))    &   𝑄 = (ℂfld ↾s ℚ)    &    + = (+g‘𝑃)    &    · = (.r‘𝑃)    &    ↑ = (.g‘(mulGrp‘𝑃))    &   𝑃 = (Poly1‘𝑄)    &   𝐾 = (algSc‘𝑃)    &   𝑋 = (var1‘𝑄)    &   𝐷 = (deg1‘𝑄)    &   𝐹 = ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))    &   𝑀 = (ℂfld minPoly ℚ)    ⇒   (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3)
 
14-Nov-2025cos9thpiminplylem6 33821 Evaluation of the polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)). (Contributed by Thierry Arnoux, 14-Nov-2025.)
𝑂 = (exp‘((i · (2 · π)) / 3))    &   𝑍 = (𝑂↑𝑐(1 / 3))    &   𝐴 = (𝑍 + (1 / 𝑍))    &   𝑄 = (ℂfld ↾s ℚ)    &    + = (+g‘𝑃)    &    · = (.r‘𝑃)    &    ↑ = (.g‘(mulGrp‘𝑃))    &   𝑃 = (Poly1‘𝑄)    &   𝐾 = (algSc‘𝑃)    &   𝑋 = (var1‘𝑄)    &   𝐷 = (deg1‘𝑄)    &   𝐹 = ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))    &   (𝜑 → 𝑌 ∈ ℂ)    ⇒   (𝜑 → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑌) = ((𝑌↑3) + ((-3 · 𝑌) + 1)))
 
14-Nov-2025cos9thpiminplylem5 33820 The constructed complex number 𝐴 is a root of the polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)). (Contributed by Thierry Arnoux, 14-Nov-2025.)
𝑂 = (exp‘((i · (2 · π)) / 3))    &   𝑍 = (𝑂↑𝑐(1 / 3))    &   𝐴 = (𝑍 + (1 / 𝑍))    ⇒   ((𝐴↑3) + ((-3 · 𝐴) + 1)) = 0
 
14-Nov-2025cos9thpiminplylem4 33819 Lemma for cos9thpiminply 33822. (Contributed by Thierry Arnoux, 14-Nov-2025.)
𝑂 = (exp‘((i · (2 · π)) / 3))    &   𝑍 = (𝑂↑𝑐(1 / 3))    ⇒   ((𝑍↑6) + (𝑍↑3)) = -1
 
14-Nov-2025cos9thpiminplylem3 33818 Lemma for cos9thpiminply 33822. (Contributed by Thierry Arnoux, 14-Nov-2025.)
𝑂 = (exp‘((i · (2 · π)) / 3))    ⇒   ((𝑂↑2) + (𝑂 + 1)) = 0
 
14-Nov-2025vr1nz 33561 A univariate polynomial variable cannot be the zero polynomial. (Contributed by Thierry Arnoux, 14-Nov-2025.)
𝑋 = (var1‘𝑈)    &   𝑍 = (0g‘𝑃)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝑃 = (Poly1‘𝑈)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑆 ∈ NzRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    ⇒   (𝜑 → 𝑋 ≠ 𝑍)
 
14-Nov-2025ressply1evls1 33535 Subring evaluation of a univariate polynomial is the same as the subring evaluation in the bigger ring. (Contributed by Thierry Arnoux, 14-Nov-2025.)
𝐺 = (𝐸 ↾s 𝑅)    &   𝑂 = (𝐸 evalSub1 𝑆)    &   𝑄 = (𝐺 evalSub1 𝑆)    &   𝑃 = (Poly1‘𝐾)    &   𝐾 = (𝐸 ↾s 𝑆)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → 𝐸 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝐸))    &   (𝜑 → 𝑆 ∈ (SubRing‘𝐺))    &   (𝜑 → 𝐹 ∈ 𝐵)    ⇒   (𝜑 → (𝑄‘𝐹) = ((𝑂‘𝐹) ↾ 𝑅))
 
14-Nov-2025efne0 16007 The exponential of a complex number is nonzero. Corollary 15-4.3 of [Gleason] p. 309. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.) (Proof shortened by TA, 14-Nov-2025.)
(𝐴 ∈ ℂ → (exp‘𝐴) ≠ 0)
 
14-Nov-2025modaddid 13816 The sums of two nonnegative integers less than the modulus and an integer are equal iff the two nonnegative integers are equal. (Contributed by AV, 14-Nov-2025.)
𝐼 = (0..^𝑁)    ⇒   ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) ∧ 𝐾 ∈ ℤ) → (((𝑋 + 𝐾) mod 𝑁) = ((𝑌 + 𝐾) mod 𝑁) ↔ 𝑋 = 𝑌))
 
14-Nov-2025modaddb 13815 Addition property of the modulo operation. Biconditional version of modadd1 13814 by applying modadd1 13814 twice. (Contributed by AV, 14-Nov-2025.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ+)) → ((𝐴 mod 𝐷) = (𝐵 mod 𝐷) ↔ ((𝐴 + 𝐶) mod 𝐷) = ((𝐵 + 𝐶) mod 𝐷)))
 
13-Nov-2025iscmd 49791 The universal property of colimits of a diagram. (Contributed by Zhi Wang, 13-Nov-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   𝐴 = (Base‘𝐶)    &   𝑁 = (𝐷 Nat 𝐶)    &   𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    ⇒   (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑅 ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))
 
13-Nov-2025coccom 49789 A co-cone to a diagram commutes with the diagram. (Contributed by Zhi Wang, 13-Nov-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   𝐴 = (Base‘𝐶)    &   𝑁 = (𝐷 Nat 𝐶)    &   𝐵 = (Base‘𝐷)    &   𝐾 = ((1st ‘𝐿)‘𝑋)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑀 ∈ (𝑌𝐽𝑍))    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐶)    &   (𝜑 → 𝑅 ∈ (𝐹𝑁𝐾))    ⇒   (𝜑 → (𝑅‘𝑌) = ((𝑅‘𝑍)(⟨((1st ‘𝐹)‘𝑌), ((1st ‘𝐹)‘𝑍)⟩ · 𝑋)((𝑌(2nd ‘𝐹)𝑍)‘𝑀)))
 
13-Nov-2025concom 49788 A cone to a diagram commutes with the diagram. (Contributed by Zhi Wang, 13-Nov-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   𝐴 = (Base‘𝐶)    &   𝑁 = (𝐷 Nat 𝐶)    &   𝐵 = (Base‘𝐷)    &   𝐾 = ((1st ‘𝐿)‘𝑋)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑀 ∈ (𝑌𝐽𝑍))    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐶)    &   (𝜑 → 𝑅 ∈ (𝐾𝑁𝐹))    ⇒   (𝜑 → (𝑅‘𝑍) = (((𝑌(2nd ‘𝐹)𝑍)‘𝑀)(⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍))(𝑅‘𝑌)))
 
13-Nov-2025coccl 49787 A natural transformation to a constant functor of an object maps to morphisms whose codomain is the object. Therefore, the range of the second component of a co-cone are morphisms with a common codomain. (Contributed by Zhi Wang, 13-Nov-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   𝐴 = (Base‘𝐶)    &   𝑁 = (𝐷 Nat 𝐶)    &   𝐵 = (Base‘𝐷)    &   𝐾 = ((1st ‘𝐿)‘𝑋)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝑅 ∈ (𝐹𝑁𝐾))    ⇒   (𝜑 → (𝑅‘𝑌) ∈ (((1st ‘𝐹)‘𝑌)𝐻𝑋))
 
13-Nov-2025concl 49786 A natural transformation from a constant functor of an object maps to morphisms whose domain is the object. Therefore, the range of the second component of a cone are morphisms with a common domain. (Contributed by Zhi Wang, 13-Nov-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   𝐴 = (Base‘𝐶)    &   𝑁 = (𝐷 Nat 𝐶)    &   𝐵 = (Base‘𝐷)    &   𝐾 = ((1st ‘𝐿)‘𝑋)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝑅 ∈ (𝐾𝑁𝐹))    ⇒   (𝜑 → (𝑅‘𝑌) ∈ (𝑋𝐻((1st ‘𝐹)‘𝑌)))
 
13-Nov-2025relcmd 49785 The set of colimits of a diagram is a relation. (Contributed by Zhi Wang, 13-Nov-2025.)
Rel ((𝐶 Colimit 𝐷)‘𝐹)
 
13-Nov-2025reldmcmd2 49779 The domain of (𝐶 Colimit 𝐷) is a relation. (Contributed by Zhi Wang, 13-Nov-2025.)
Rel dom (𝐶 Colimit 𝐷)
 
13-Nov-2025oppfval2 49262 Value of the opposite functor. (Contributed by Zhi Wang, 13-Nov-2025.)
(𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩)
 
13-Nov-2025oppfvallem 49260 Lemma for oppfval 49261. (Contributed by Zhi Wang, 13-Nov-2025.)
(𝐹(𝐶 Func 𝐷)𝐺 → (Rel 𝐺 ∧ Rel dom 𝐺))
 
13-Nov-2025oppfvalg 49251 Value of the opposite functor. (Contributed by Zhi Wang, 13-Nov-2025.)
((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 oppFunc 𝐺) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
 
13-Nov-2025reldmoppf 49250 The domain of oppFunc is a relation. (Contributed by Zhi Wang, 13-Nov-2025.)
Rel dom oppFunc
 
13-Nov-2025df-oppf 49248 Definition of the operation generating opposite functors. Definition 3.41 of [Adamek] p. 39. The object part of the functor is unchanged while the morphism part is transposed due to reversed direction of arrows in the opposite category. The opposite functor is a functor on opposite categories (oppfoppc 49266). (Contributed by Zhi Wang, 4-Nov-2025.) Better reverse closure. (Revised by Zhi Wang, 13-Nov-2025.)
oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
 
13-Nov-2025lamberte 47012 A value of Lambert W (product logarithm) function at e. (Contributed by Ender Ting, 13-Nov-2025.)
𝑅 = ◡(𝑥 ∈ ℂ ↦ (𝑥 · (exp‘𝑥)))    ⇒   e𝑅1
 
13-Nov-2025lambert0 47011 A value of Lambert W (product logarithm) function at zero. (Contributed by Ender Ting, 13-Nov-2025.)
𝑅 = ◡(𝑥 ∈ ℂ ↦ (𝑥 · (exp‘𝑥)))    ⇒   0𝑅0
 
13-Nov-2025sbralie 3319 Implicit to explicit substitution that swaps variables in a restrictedly universally quantified expression. (Contributed by NM, 5-Sep-2004.) Avoid ax-ext 2705, df-cleq 2725, df-clel 2808. (Revised by Wolf Lammen, 10-Mar-2025.) Avoid ax-10 2146, ax-12 2182. (Revised by SN, 13-Nov-2025.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
 
12-Nov-2025cmdfval2 49781 The set of colimits of a diagram. (Contributed by Zhi Wang, 12-Nov-2025.)
((𝐶 Colimit 𝐷)‘𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
 
12-Nov-2025cmdfval 49775 Function value of Colimit. (Contributed by Zhi Wang, 12-Nov-2025.)
(𝐶 Colimit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓))
 
12-Nov-2025reldmcmd 49773 The domain of Colimit is a relation. (Contributed by Zhi Wang, 12-Nov-2025.)
Rel dom Colimit
 
12-Nov-2025reldmlmd 49772 The domain of Limit is a relation. (Contributed by Zhi Wang, 12-Nov-2025.)
Rel dom Limit
 
12-Nov-2025df-cmd 49771 A co-cone (or cocone) to a diagram (see df-lmd 49770 for definition), or a natural sink for a diagram in a category 𝐶 is a pair of an object 𝑋 in 𝐶 and a natural transformation from the diagram to the constant functor (or constant diagram) of the object 𝑋. The second component associates each object in the index category with a morphism in 𝐶 whose codomain is 𝑋 (coccl 49787). The naturality guarantees that the combination of the diagram with the co-cone must commute (coccom 49789). Definition 11.27(1) of [Adamek] p. 202.

A colimit of a diagram 𝐹:𝐷⟶𝐶 of type 𝐷 in category 𝐶 is a universal pair from the diagram to the diagonal functor (𝐶Δfunc𝐷). The universal pair is a co-cone to the diagram satisfying the universal property, that each co-cone to the diagram uniquely factors through the colimit. (iscmd 49791). Definition 11.27(2) of [Adamek] p. 202.

Initial objects (initocmd 49794), coproducts, coequalizers, pushouts, and direct limits can be considered as colimits of some diagram; colimits can be further generalized as left Kan extensions (cmdlan 49797).

"cmd" is short for "colimit of a diagram". See df-lmd 49770 for the dual concept (lmddu 49792, cmddu 49793). (Contributed by Zhi Wang, 12-Nov-2025.)

Colimit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)))
 
12-Nov-2025df-lmd 49770 A diagram of type 𝐷 or a 𝐷-shaped diagram in a category 𝐶, is a functor 𝐹:𝐷⟶𝐶 where the source category 𝐷, usually small or even finite, is called the index category or the scheme of the diagram. The actual objects and morphisms in 𝐷 are largely irrelevant; only the way in which they are interrelated matters. The diagram is thought of as indexing a collection of objects and morphisms in 𝐶 patterned on 𝐷. Definition 11.1(1) of [Adamek] p. 193.

A cone to a diagram, or a natural source for a diagram in a category 𝐶 is a pair of an object 𝑋 in 𝐶 and a natural transformation from the constant functor (or constant diagram) of the object 𝑋 to the diagram. The second component associates each object in the index category with a morphism in 𝐶 whose domain is 𝑋 (concl 49786). The naturality guarantees that the combination of the diagram with the cone must commute (concom 49788). Definition 11.3(1) of [Adamek] p. 193.

A limit of a diagram 𝐹:𝐷⟶𝐶 of type 𝐷 in category 𝐶 is a universal pair from the diagonal functor (𝐶Δfunc𝐷) to the diagram. The universal pair is a cone to the diagram satisfying the universal property, that each cone to the diagram uniquely factors through the limit (islmd 49790). Definition 11.3(2) of [Adamek] p. 194.

Terminal objects (termolmd 49795), products, equalizers, pullbacks, and inverse limits can be considered as limits of some diagram; limits can be further generalized as right Kan extensions (lmdran 49796).

"lmd" is short for "limit of a diagram". See df-cmd 49771 for the dual concept (lmddu 49792, cmddu 49793). (Contributed by Zhi Wang, 12-Nov-2025.)

Limit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ (( oppFunc ‘(𝑐Δfunc𝑑))((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐)))𝑓)))
 
12-Nov-2025upfval 49301 Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.) (Proof shortened by Zhi Wang, 12-Nov-2025.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   𝑂 = (comp‘𝐸)    ⇒   (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})
 
12-Nov-2025reldmfunc 49200 The domain of Func is a relation. (Contributed by Zhi Wang, 12-Nov-2025.)
Rel dom Func
 
11-Nov-2025discthing 49586 A discrete category, i.e., a category where all morphisms are identity morphisms, is thin. Example 3.26(1) of [Adamek] p. 33. (Contributed by Zhi Wang, 11-Nov-2025.)
(𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝐶 ∈ Cat)    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐻𝑦) = if(𝑥 = 𝑦, {𝐼}, ∅))    ⇒   (𝜑 → 𝐶 ∈ ThinCat)
 
11-Nov-2025indcthing 49585 An indiscrete category, i.e., a category where all hom-sets have exactly one morphism, is thin. (Contributed by Zhi Wang, 11-Nov-2025.)
(𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝐶 ∈ Cat)    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐻𝑦) = {𝐹})    ⇒   (𝜑 → 𝐶 ∈ ThinCat)
 
11-Nov-2025idfullsubc 49286 The source category of an inclusion functor is a full subcategory of the target category if the inclusion functor is full. Remark 4.4(2) in [Adamek] p. 49. See also ressffth 17849. (Contributed by Zhi Wang, 11-Nov-2025.)
𝐼 = (idfunc‘𝐶)    &   𝐻 = (Homf ‘𝐷)    &   𝐽 = (Homf ‘𝐸)    &   𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    ⇒   (𝐼 ∈ (𝐷 Full 𝐸) → (𝐵 ⊆ 𝐶 ∧ (𝐽 ↾ (𝐵 × 𝐵)) = 𝐻))
 
11-Nov-2025gpgedgiov 48189 The edges of the generalized Petersen graph GPG(N,K) between an inside and an outside vertex. (Contributed by AV, 11-Nov-2025.)
𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   𝐼 = (0..^𝑁)    &   𝐺 = (𝑁 gPetersenGr 𝐾)    &   𝐸 = (Edg‘𝐺)    ⇒   (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸 ↔ 𝑋 = 𝑌))
 
11-Nov-2025pw2cutp1 28382 Simplify pw2cut 28381 in the case of successors of surreal integers. (Contributed by Scott Fenton, 11-Nov-2025.)
(𝜑 → 𝐴 ∈ ℤs)    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))}) = (((2s ·s 𝐴) +s 1s ) /su (2s↑s(𝑁 +s 1s ))))
 
10-Nov-2025idsubc 49285 The source category of an inclusion functor is a subcategory of the target category. See also Remark 4.4 in [Adamek] p. 49. (Contributed by Zhi Wang, 10-Nov-2025.)
𝐼 = (idfunc‘𝐶)    &   𝐻 = (Homf ‘𝐷)    ⇒   (𝐼 ∈ (𝐷 Func 𝐸) → 𝐻 ∈ (Subcat‘𝐸))
 
10-Nov-2025idfth 49283 The inclusion functor is a faithful functor. (Contributed by Zhi Wang, 10-Nov-2025.)
𝐼 = (idfunc‘𝐶)    ⇒   (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸))
 
10-Nov-2025fthcomf 49282 Source categories of a faithful functor have the same base, hom-sets and composition operation if the composition is compatible in images of the functor. (Contributed by Zhi Wang, 10-Nov-2025.)
(𝜑 → 𝐹(𝐴 Faith 𝐶)𝐺)    &   (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)    &   (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐶)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)))    ⇒   (𝜑 → (compf‘𝐴) = (compf‘𝐵))
 
10-Nov-2025imaidfu2 49236 The image of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)
𝐼 = (idfunc‘𝐶)    &   (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Homf ‘𝐷)    &   𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝)))    &   (𝜑 → 𝑆 = (Base‘𝐷))    ⇒   (𝜑 → 𝐽 = 𝐾)
 
10-Nov-2025imaidfu 49235 The image of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)
𝐼 = (idfunc‘𝐶)    &   (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Homf ‘𝐷)    &   𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝)))    &   𝑆 = ((1st ‘𝐼) “ 𝐴)    ⇒   (𝜑 → (𝐽 ↾ (𝑆 × 𝑆)) = 𝐾)
 
10-Nov-2025imaidfu2lem 49234 Lemma for imaidfu2 49236. (Contributed by Zhi Wang, 10-Nov-2025.)
𝐼 = (idfunc‘𝐶)    &   (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    ⇒   (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
 
10-Nov-2025idfu2nda 49228 Value of the morphism part of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)
𝐼 = (idfunc‘𝐶)    &   (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐵 = (Base‘𝐷))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐻 = (𝑋(Hom ‘𝐷)𝑌))    ⇒   (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ 𝐻))
 
10-Nov-2025idfu1a 49227 Value of the object part of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)
𝐼 = (idfunc‘𝐶)    &   (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐵 = (Base‘𝐷))    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋)
 
10-Nov-2025idfu1sta 49226 Value of the object part of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)
𝐼 = (idfunc‘𝐶)    &   (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐵 = (Base‘𝐷))    ⇒   (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵))
 
10-Nov-2025idfu1stalem 49225 Lemma for idfu1sta 49226. (Contributed by Zhi Wang, 10-Nov-2025.)
𝐼 = (idfunc‘𝐶)    &   (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐵 = (Base‘𝐷))    ⇒   (𝜑 → 𝐵 = (Base‘𝐶))
 
10-Nov-2025idfurcl 49223 Reverse closure for an identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)
((idfunc‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
 
10-Nov-2025funchomf 49222 Source categories of a functor have the same set of objects and morphisms. (Contributed by Zhi Wang, 10-Nov-2025.)
(𝜑 → 𝐹(𝐴 Func 𝐶)𝐺)    &   (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)    ⇒   (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
 
10-Nov-2025pgjsgr 48216 A Petersen graph is a simple graph. (Contributed by AV, 10-Nov-2025.)
(5 gPetersenGr 2) ∈ USGraph
 
10-Nov-2025pglem 48215 Lemma for theorems about Petersen graphs. (Contributed by AV, 10-Nov-2025.)
2 ∈ (1..^(⌈‘(5 / 2)))
 
10-Nov-2025ndmfvrcl 6861 Reverse closure law for function with the empty set not in its domain (if 𝑅 = 𝑆). (Contributed by NM, 26-Apr-1996.) The class containing the function value does not have to be the domain. (Revised by Zhi Wang, 10-Nov-2025.)
dom 𝐹 = 𝑆    &    ¬ ∅ ∈ 𝑅    ⇒   ((𝐹‘𝐴) ∈ 𝑅 → 𝐴 ∈ 𝑆)
 
9-Nov-2025pgn4cyclex 48250 A cycle in a Petersen graph G(5,2) does not have length 4. (Contributed by AV, 9-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    ⇒   (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ≠ 4)
 
9-Nov-2025pgnbgreunbgr 48249 In a Petersen graph, two different neighbors of a vertex have exactly one common neighbor, which is the vertex itself. (Contributed by AV, 9-Nov-2025.)
𝐺 = (5 gPetersenGr 2)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) → ∃!𝑥 ∈ 𝑉 {{𝐾, 𝑥}, {𝑥, 𝐿}} ⊆ 𝐸)
 
9-Nov-2025cos9thpiminplylem2 33817 The polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)) has no rational roots. (Contributed by Thierry Arnoux, 9-Nov-2025.)
(𝜑 → 𝑋 ∈ ℚ)    ⇒   (𝜑 → ((𝑋↑3) + ((-3 · 𝑋) + 1)) ≠ 0)
 
9-Nov-2025cos9thpiminplylem1 33816 The polynomial ((𝑋↑3) + ((-3 · (𝑋↑2)) + 1)) has no integer roots. (Contributed by Thierry Arnoux, 9-Nov-2025.)
(𝜑 → 𝑋 ∈ ℤ)    ⇒   (𝜑 → ((𝑋↑3) + ((-3 · (𝑋↑2)) + 1)) ≠ 0)
 
9-Nov-2025oexpled 32835 Odd power monomials are monotonic. (Contributed by Thierry Arnoux, 9-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝑁)    &   (𝜑 → 𝐴 ≤ 𝐵)    ⇒   (𝜑 → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
 
9-Nov-2025expevenpos 32834 Even powers are positive. (Contributed by Thierry Arnoux, 9-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 2 ∥ 𝑁)    ⇒   (𝜑 → 0 ≤ (𝐴↑𝑁))
 
9-Nov-2025elq2 32799 Elementhood in the rational numbers, providing the canonical representation. (Contributed by Thierry Arnoux, 9-Nov-2025.)
(𝑄 ∈ ℚ → ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ (𝑄 = (𝑝 / 𝑞) ∧ (𝑝 gcd 𝑞) = 1))
 
9-Nov-2025receqid 32732 Real numbers equal to their own reciprocal have absolute value 1. (Contributed by Thierry Arnoux, 9-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 ≠ 0)    ⇒   (𝜑 → ((1 / 𝐴) = 𝐴 ↔ (abs‘𝐴) = 1))
 
9-Nov-2025sgnval2 32722 Value of the signum of a real number, expresssed using absolute value. (Contributed by Thierry Arnoux, 9-Nov-2025.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (sgn‘𝐴) = (𝐴 / (abs‘𝐴)))
 
9-Nov-2025zs12negsclb 28392 A surreal is a dyadic fraction iff its negative is. (Contributed by Scott Fenton, 9-Nov-2025.)
(𝐴 ∈ No → (𝐴 ∈ ℤs[1/2] ↔ ( -us ‘𝐴) ∈ ℤs[1/2]))
 
9-Nov-2025zs12negscl 28389 The dyadics are closed under negation. (Contributed by Scott Fenton, 9-Nov-2025.)
(𝐴 ∈ ℤs[1/2] → ( -us ‘𝐴) ∈ ℤs[1/2])
 
8-Nov-2025gpgedgel 48174 An edge in a generalized Petersen graph 𝐺. (Contributed by AV, 29-Aug-2025.) (Proof shortened by AV, 8-Nov-2025.)
𝐼 = (0..^𝑁)    &   𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   𝐺 = (𝑁 gPetersenGr 𝐾)    &   𝐸 = (Edg‘𝐺)    ⇒   ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑌 ∈ 𝐸 ↔ ∃𝑥 ∈ 𝐼 (𝑌 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑌 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑌 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
 
8-Nov-2025zs12ge0 28394 An expression for non-negative dyadic rationals. (Contributed by Scott Fenton, 8-Nov-2025.)
((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
 
8-Nov-2025pw2divsnegd 28373 Move negative sign inside of a power of two division. (Contributed by Scott Fenton, 8-Nov-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → ( -us ‘(𝐴 /su (2s↑s𝑁))) = (( -us ‘𝐴) /su (2s↑s𝑁)))
 
8-Nov-2025nnexpscl 28357 Closure law for positive surreal integer exponentiation. (Contributed by Scott Fenton, 8-Nov-2025.)
((𝐴 ∈ ℕs ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ ℕs)
 
8-Nov-2025eucliddivs 28302 Euclid's division lemma for surreal numbers. (Contributed by Scott Fenton, 8-Nov-2025.)
((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝐴 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵))
 
8-Nov-2025nnm1n0s 28301 A positive surreal integer minus one is a non-negative surreal integer. (Contributed by Scott Fenton, 8-Nov-2025.)
(𝑁 ∈ ℕs → (𝑁 -s 1s ) ∈ ℕ0s)
 
8-Nov-2025nn1m1nns 28300 Every positive surreal integer is either one or a successor. (Contributed by Scott Fenton, 8-Nov-2025.)
(𝐴 ∈ ℕs → (𝐴 = 1s ∨ (𝐴 -s 1s ) ∈ ℕs))
 
8-Nov-2025n0slem1lt 28294 Non-negative surreal ordering relation. (Contributed by Scott Fenton, 8-Nov-2025.)
((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ (𝑀 -s 1s ) <s 𝑁))
 
8-Nov-2025onsiso 28206 The birthday function restricted to the surreal ordinals forms an order-preserving isomorphism with the regular ordinals. (Contributed by Scott Fenton, 8-Nov-2025.)
( bday ↾ Ons) Isom <s , E (Ons, On)
 
7-Nov-2025imasubc3 49281 An image of a functor injective on objects is a subcategory. Remark 4.2(3) of [Adamek] p. 48. (Contributed by Zhi Wang, 7-Nov-2025.)
𝑆 = (𝐹 “ 𝐴)    &   𝐻 = (Hom ‘𝐷)    &   𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    &   (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑 → Fun ◡𝐹)    ⇒   (𝜑 → 𝐾 ∈ (Subcat‘𝐸))
 
7-Nov-2025imaf1co 49280 An image of a functor whose object part is injective preserves the composition. (Contributed by Zhi Wang, 7-Nov-2025.)
𝑆 = (𝐹 “ 𝐴)    &   𝐻 = (Hom ‘𝐷)    &   𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    &   (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &    ∙ = (comp‘𝐸)    &   (𝜑 → 𝐹:𝐵–1-1→𝐶)    &   (𝜑 → 𝑋 ∈ 𝑆)    &   (𝜑 → 𝑌 ∈ 𝑆)    &   (𝜑 → 𝑍 ∈ 𝑆)    &   (𝜑 → 𝑀 ∈ (𝑋𝐾𝑌))    &   (𝜑 → 𝑁 ∈ (𝑌𝐾𝑍))    ⇒   (𝜑 → (𝑁(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝑀) ∈ (𝑋𝐾𝑍))
 
7-Nov-2025imaid 49279 An image of a functor preserves the identity morphism. (Contributed by Zhi Wang, 7-Nov-2025.)
𝑆 = (𝐹 “ 𝐴)    &   𝐻 = (Hom ‘𝐷)    &   𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    &   (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   𝐼 = (Id‘𝐸)    &   (𝜑 → 𝑋 ∈ 𝑆)    ⇒   (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐾𝑋))
 
7-Nov-2025imassc 49278 An image of a functor satisfies the subcategory subset relation. (Contributed by Zhi Wang, 7-Nov-2025.)
𝑆 = (𝐹 “ 𝐴)    &   𝐻 = (Hom ‘𝐷)    &   𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    &   (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   𝐽 = (Homf ‘𝐸)    ⇒   (𝜑 → 𝐾 ⊆cat 𝐽)
 
7-Nov-2025imasubc2 49277 An image of a full functor is a (full) subcategory. Remark 4.2(3) of [Adamek] p. 48. (Contributed by Zhi Wang, 7-Nov-2025.)
𝑆 = (𝐹 “ 𝐴)    &   𝐻 = (Hom ‘𝐷)    &   𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    &   (𝜑 → 𝐹(𝐷 Full 𝐸)𝐺)    ⇒   (𝜑 → 𝐾 ∈ (Subcat‘𝐸))
 
7-Nov-2025imasubc 49276 An image of a full functor is a full subcategory. Remark 4.2(3) of [Adamek] p. 48. (Contributed by Zhi Wang, 7-Nov-2025.)
𝑆 = (𝐹 “ 𝐴)    &   𝐻 = (Hom ‘𝐷)    &   𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    &   (𝜑 → 𝐹(𝐷 Full 𝐸)𝐺)    &   𝐶 = (Base‘𝐸)    &   𝐽 = (Homf ‘𝐸)    ⇒   (𝜑 → (𝐾 Fn (𝑆 × 𝑆) ∧ 𝑆 ⊆ 𝐶 ∧ (𝐽 ↾ (𝑆 × 𝑆)) = 𝐾))
 
7-Nov-2025imaf1hom 49233 The hom-set of an image of a functor injective on objects. (Contributed by Zhi Wang, 7-Nov-2025.)
𝑆 = (𝐹 “ 𝐴)    &   (𝜑 → 𝐹:𝐵–1-1→𝐶)    &   (𝜑 → 𝑋 ∈ 𝑆)    &   (𝜑 → 𝑌 ∈ 𝑆)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    ⇒   (𝜑 → (𝑋𝐾𝑌) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))
 
7-Nov-2025imaf1homlem 49232 Lemma for imaf1hom 49233 and other theorems. (Contributed by Zhi Wang, 7-Nov-2025.)
𝑆 = (𝐹 “ 𝐴)    &   (𝜑 → 𝐹:𝐵–1-1→𝐶)    &   (𝜑 → 𝑋 ∈ 𝑆)    ⇒   (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))
 
7-Nov-2025imasubclem3 49231 Lemma for imasubc 49276. (Contributed by Zhi Wang, 7-Nov-2025.)
(𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ 𝑊)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐾 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ∪ 𝑧 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐺 “ {𝑦}))((𝐻‘𝐶) “ 𝐷))    ⇒   (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷))
 
7-Nov-2025imasubclem2 49230 Lemma for imasubc 49276. (Contributed by Zhi Wang, 7-Nov-2025.)
(𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ 𝑊)    &   𝐾 = (𝑦 ∈ 𝑋, 𝑧 ∈ 𝑌 ↦ ∪ 𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷))    ⇒   (𝜑 → 𝐾 Fn (𝑋 × 𝑌))
 
7-Nov-2025inisegn0a 48960 The inverse image of a singleton subset of an image is non-empty. (Contributed by Zhi Wang, 7-Nov-2025.)
(𝐴 ∈ (𝐹 “ 𝐵) → (◡𝐹 “ {𝐴}) ≠ ∅)
 
7-Nov-2025pw2divsdird 28372 Distribution of surreal division over addition for powers of two. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → ((𝐴 +s 𝐵) /su (2s↑s𝑁)) = ((𝐴 /su (2s↑s𝑁)) +s (𝐵 /su (2s↑s𝑁))))
 
7-Nov-2025pw2divsrecd 28371 Relationship between surreal division and reciprocal for powers of two. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → (𝐴 /su (2s↑s𝑁)) = (𝐴 ·s ( 1s /su (2s↑s𝑁))))
 
7-Nov-2025pw2ge0divsd 28370 Divison of a non-negative surreal by a power of two. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → ( 0s ≤s 𝐴 ↔ 0s ≤s (𝐴 /su (2s↑s𝑁))))
 
7-Nov-2025pw2gt0divsd 28369 Division of a positive surreal by a power of two. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → ( 0s <s 𝐴 ↔ 0s <s (𝐴 /su (2s↑s𝑁))))
 
7-Nov-2025pw2divscan2d 28366 A cancellation law for surreal division by powers of two. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → ((2s↑s𝑁) ·s (𝐴 /su (2s↑s𝑁))) = 𝐴)
 
7-Nov-2025pw2divscan3d 28365 Cancellation law for surreal division by powers of two. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → (((2s↑s𝑁) ·s 𝐴) /su (2s↑s𝑁)) = 𝐴)
 
7-Nov-2025pw2divsmuld 28364 Relationship between surreal division and multiplication for powers of two. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → ((𝐴 /su (2s↑s𝑁)) = 𝐵 ↔ ((2s↑s𝑁) ·s 𝐵) = 𝐴))
 
7-Nov-2025pw2divscld 28363 Division closure for powers of two. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   (𝜑 → (𝐴 /su (2s↑s𝑁)) ∈ No )
 
7-Nov-2025expadds 28359 Sum of exponents law for surreals. (Contributed by Scott Fenton, 7-Nov-2025.)
((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁)))
 
7-Nov-2025n0expscl 28356 Closure law for non-negative surreal integer exponentiation. (Contributed by Scott Fenton, 7-Nov-2025.)
((𝐴 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ ℕ0s)
 
7-Nov-2025expscllem 28354 Lemma for proving non-negative surreal integer exponentiation closure. (Contributed by Scott Fenton, 7-Nov-2025.)
𝐹 ⊆ No     &   ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ·s 𝑦) ∈ 𝐹)    &    1s ∈ 𝐹    ⇒   ((𝐴 ∈ 𝐹 ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ 𝐹)
 
7-Nov-2025bdayn0sf1o 28296 The birthday function restricted to the non-negative surreal integers is a bijection with the finite ordinals. (Contributed by Scott Fenton, 7-Nov-2025.)
( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
 
7-Nov-2025bdayn0p1 28295 The birthday of 𝐴 +s 1s is the successor of the birthday of 𝐴 when 𝐴 is a non-negative surreal integer. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝐴 ∈ ℕ0s → ( bday ‘(𝐴 +s 1s )) = suc ( bday ‘𝐴))
 
7-Nov-2025n0sleltp1 28293 Non-negative surreal ordering relation. (Contributed by Scott Fenton, 7-Nov-2025.)
((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ 𝑀 <s (𝑁 +s 1s )))
 
7-Nov-2025n0sltp1le 28292 Non-negative surreal ordering relation. (Contributed by Scott Fenton, 7-Nov-2025.)
((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑀 +s 1s ) ≤s 𝑁))
 
7-Nov-2025n0subs2 28291 Subtraction of non-negative surreal integers. (Contributed by Scott Fenton, 7-Nov-2025.)
((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕs))
 
7-Nov-2025n0cutlt 28286 A non-negative surreal integer is the simplest number greater than all previous non-negative surreal integers. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝐴 ∈ ℕ0s → 𝐴 = ({𝑥 ∈ ℕ0s ∣ 𝑥 <s 𝐴} |s ∅))
 
7-Nov-2025onltn0s 28285 A surreal ordinal that is less than a non-negative integer is a non-negative integer. (Contributed by Scott Fenton, 7-Nov-2025.)
((𝐴 ∈ Ons ∧ 𝐵 ∈ ℕ0s ∧ 𝐴 <s 𝐵) → 𝐴 ∈ ℕ0s)
 
7-Nov-2025n0scut2 28264 A cut form for the successor of a non-negative surreal integer. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝐴 ∈ ℕ0s → (𝐴 +s 1s ) = ({𝐴} |s ∅))
 
7-Nov-2025bdayon 28210 The birthday of a surreal ordinal is the set of all previous ordinal birthdays. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝐴 ∈ Ons → ( bday ‘𝐴) = ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}))
 
7-Nov-2025onslt 28205 Less-than is the same as birthday comparison over surreal ordinals. (Contributed by Scott Fenton, 7-Nov-2025.)
((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 <s 𝐵 ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
 
7-Nov-2025onnolt 28204 If a surreal ordinal is less than a given surreal, then it is simpler. (Contributed by Scott Fenton, 7-Nov-2025.)
((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → ( bday ‘𝐴) ∈ ( bday ‘𝐵))
 
7-Nov-2025subseq0d 28045 The difference between two surreals is zero iff they are equal. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → ((𝐴 -s 𝐵) = 0s ↔ 𝐴 = 𝐵))
 
7-Nov-2025subscan2d 28044 Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 -s 𝐶) = (𝐵 -s 𝐶) ↔ 𝐴 = 𝐵))
 
7-Nov-2025subscan1d 28043 Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐶 -s 𝐴) = (𝐶 -s 𝐵) ↔ 𝐴 = 𝐵))
 
7-Nov-2025newbdayim 27849 One direction of the biconditional in newbday 27848. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝑋 ∈ ( N ‘𝐴) → ( bday ‘𝑋) = 𝐴)
 
7-Nov-2025rightgt 27810 A member of a surreal's right set is greater than it. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝐴 ∈ ( R ‘𝐵) → 𝐵 <s 𝐴)
 
7-Nov-2025leftlt 27809 A member of a surreal's left set is less than it. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝐴 ∈ ( L ‘𝐵) → 𝐴 <s 𝐵)
 
7-Nov-2025elright 27808 Membership in the right set of a surreal. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝐴 ∈ ( R ‘𝐵) ↔ (𝐴 ∈ ( O ‘( bday ‘𝐵)) ∧ 𝐵 <s 𝐴))
 
7-Nov-2025elleft 27807 Membership in the left set of a surreal. (Contributed by Scott Fenton, 7-Nov-2025.)
(𝐴 ∈ ( L ‘𝐵) ↔ (𝐴 ∈ ( O ‘( bday ‘𝐵)) ∧ 𝐴 <s 𝐵))
 
6-Nov-2025cnelsubc 49729 Remark 4.2(2) of [Adamek] p. 48. There exists a category satisfying all conditions for a subcategory but the compatibility of identity morphisms. Therefore such condition in df-subc 17721 is necessary. A stronger statement than nelsubc3 49196. (Contributed by Zhi Wang, 6-Nov-2025.)
∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ ¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ (𝑐 ↾cat 𝑗) ∈ Cat))
 
6-Nov-2025cnelsubclem 49728 Lemma for cnelsubc 49729. (Contributed by Zhi Wang, 6-Nov-2025.)
𝐽 ∈ V    &   𝑆 ∈ V    &   (𝐶 ∈ Cat ∧ 𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat (Homf ‘𝐶) ∧ ¬ ∀𝑥 ∈ 𝑆 ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐶 ↾cat 𝐽) ∈ Cat))    ⇒   ∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ ¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ (𝑐 ↾cat 𝑗) ∈ Cat))
 
6-Nov-2025setc1onsubc 49727 Construct a category with one object and two morphisms and prove that category (SetCat‘1o) satisfies all conditions for a subcategory but the compatibility of identity morphisms, showing the necessity of the latter condition in defining a subcategory. Exercise 4A of [Adamek] p. 58. (Contributed by Zhi Wang, 6-Nov-2025.)
𝐶 = {⟨(Base‘ndx), {∅}⟩, ⟨(Hom ‘ndx), {⟨∅, ∅, 2o⟩}⟩, ⟨(comp‘ndx), {⟨⟨∅, ∅⟩, ∅, · ⟩}⟩}    &    · = (𝑓 ∈ 2o, 𝑔 ∈ 2o ↦ (𝑓 ∩ 𝑔))    &   𝐸 = (SetCat‘1o)    &   𝐽 = (Homf ‘𝐸)    &   𝑆 = 1o    &   𝐻 = (Homf ‘𝐶)    &    1 = (Id‘𝐶)    &   𝐷 = (𝐶 ↾cat 𝐽)    ⇒   (𝐶 ∈ Cat ∧ 𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))
 
6-Nov-2025fulltermc2 49637 Given a full functor to a terminal category, the source category must not have empty hom-sets. (Contributed by Zhi Wang, 17-Oct-2025.) (Proof shortened by Zhi Wang, 6-Nov-2025.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐷 ∈ TermCat)    &   (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)
 
6-Nov-2025imasubclem1 49229 Lemma for imasubc 49276. (Contributed by Zhi Wang, 6-Nov-2025.)
(𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ 𝑊)    ⇒   (𝜑 → ∪ 𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷) ∈ V)
 
6-Nov-2025resccat 49199 A class 𝐶 restricted by the hom-sets of another set 𝐸, whose base is a subset of the base of 𝐶 and whose composition is compatible with 𝐶, is a category iff 𝐸 is a category. Note that the compatibility condition "resccat.1" can be weakened by removing 𝑥 ∈ 𝑆 because 𝑓 ∈ (𝑥𝐽𝑦) implies these. (Contributed by Zhi Wang, 6-Nov-2025.)
𝐷 = (𝐶 ↾cat 𝐽)    &   𝐵 = (Base‘𝐶)    &   𝑆 = (Base‘𝐸)    &   𝐽 = (Homf ‘𝐸)    &    · = (comp‘𝐶)    &    ∙ = (comp‘𝐸)    &   (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓))    &   (𝜑 → 𝐸 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   (𝜑 → (𝐷 ∈ Cat ↔ 𝐸 ∈ Cat))
 
6-Nov-2025resccatlem 49198 Lemma for resccat 49199. (Contributed by Zhi Wang, 6-Nov-2025.)
𝐷 = (𝐶 ↾cat 𝐽)    &   𝐵 = (Base‘𝐶)    &   𝑆 = (Base‘𝐸)    &   𝐽 = (Homf ‘𝐸)    &    · = (comp‘𝐶)    &    ∙ = (comp‘𝐸)    &   (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓))    &   (𝜑 → 𝐸 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐵)    &   (𝜑 → 𝐶 ∈ 𝑈)    ⇒   (𝜑 → (𝐷 ∈ Cat ↔ 𝐸 ∈ Cat))
 
6-Nov-2025ssccatid 49197 A category 𝐶 restricted by 𝐽 is a category if all of the following are satisfied: a) the base is a subset of base of 𝐶, b) all hom-sets are subsets of hom-sets of 𝐶, c) it has identity morphisms for all objects, d) the composition under 𝐶 is closed in 𝐽. But 𝐽 might not be a subcategory of 𝐶 (see cnelsubc 49729). (Contributed by Zhi Wang, 6-Nov-2025.)
𝐻 = (Homf ‘𝐶)    &   𝐷 = (𝐶 ↾cat 𝐽)    &    · = (comp‘𝐶)    &   (𝜑 → 𝐽 ⊆cat 𝐻)    &   (𝜑 → 𝐽 Fn (𝑆 × 𝑆))    &   (𝜑 → 𝐶 ∈ Cat)    &   ((𝜑 ∧ 𝑦 ∈ 𝑆) → 1 ∈ (𝑦𝐽𝑦))    &   ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ∧ 𝑚 ∈ (𝑎𝐽𝑏))) → ( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚)    &   ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ∧ 𝑚 ∈ (𝑎𝐽𝑏))) → (𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))    ⇒   (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑦 ∈ 𝑆 ↦ 1 )))
 
6-Nov-2025iineqconst2 48948 Indexed intersection of identical classes. (Contributed by Zhi Wang, 6-Nov-2025.)
((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
 
6-Nov-2025iuneqconst2 48947 Indexed union of identical classes. (Contributed by Zhi Wang, 6-Nov-2025.)
((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
 
6-Nov-2025cycldlenngric 48052 Two simple pseudographs are not isomorphic if one has a cycle and the other has no cycle of the same length. (Contributed by AV, 6-Nov-2025.)
((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ∧ ¬ ∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ¬ 𝐺 ≃𝑔𝑟 𝐻))
 
6-Nov-2025upgrimwlklen 48027 Graph isomorphisms between simple pseudographs map walks onto walks of the same length. (Contributed by AV, 6-Nov-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹(Walks‘𝐺)𝑃)    ⇒   (𝜑 → (𝐸(Walks‘𝐻)(𝑁 ∘ 𝑃) ∧ (♯‘𝐸) = (♯‘𝐹)))
 
6-Nov-2025permaxinf2 45130 The Axiom of Infinity ax-inf2 9538 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
𝐹:V–1-1-onto→V    &   𝑅 = (◡𝐹 ∘ E )    ⇒   ∃𝑥(∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ∧ ∀𝑦(𝑦𝑅𝑥 → ∃𝑧(𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))
 
6-Nov-2025permaxinf2lem 45129 Lemma for permaxinf2 45130. (Contributed by Eric Schmidt, 6-Nov-2025.)
𝐹:V–1-1-onto→V    &   𝑅 = (◡𝐹 ∘ E )    &   𝑍 = (rec((𝑣 ∈ V ↦ (◡𝐹‘((𝐹‘𝑣) ∪ {𝑣}))), (◡𝐹‘∅)) “ ω)    ⇒   ∃𝑥(∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ∧ ∀𝑦(𝑦𝑅𝑥 → ∃𝑧(𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))
 
6-Nov-2025permaxun 45128 The Axiom of Union ax-un 7674 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
𝐹:V–1-1-onto→V    &   𝑅 = (◡𝐹 ∘ E )    ⇒   ∃𝑦∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦)
 
6-Nov-2025permaxpr 45127 The Axiom of Pairing ax-pr 5372 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
𝐹:V–1-1-onto→V    &   𝑅 = (◡𝐹 ∘ E )    ⇒   ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤𝑅𝑧)
 
6-Nov-2025permaxpow 45126 The Axiom of Power Sets ax-pow 5305 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
𝐹:V–1-1-onto→V    &   𝑅 = (◡𝐹 ∘ E )    ⇒   ∃𝑦∀𝑧(∀𝑤(𝑤𝑅𝑧 → 𝑤𝑅𝑥) → 𝑧𝑅𝑦)
 
6-Nov-2025permaxnul 45125 The Null Set Axiom ax-nul 5246 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
𝐹:V–1-1-onto→V    &   𝑅 = (◡𝐹 ∘ E )    ⇒   ∃𝑥∀𝑦 ¬ 𝑦𝑅𝑥
 
6-Nov-2025permaxsep 45124 The Axiom of Separation ax-sep 5236 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148.

Note that, to prove that an instance of Separation holds in the model, 𝜑 would need have all instances of ∈ replaced with 𝑅. But this still results in an instance of this theorem, so we do establish that Separation holds. (Contributed by Eric Schmidt, 6-Nov-2025.)

𝐹:V–1-1-onto→V    &   𝑅 = (◡𝐹 ∘ E )    ⇒   ∃𝑦∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑))
 
6-Nov-2025permaxrep 45123 The Axiom of Replacement ax-rep 5219 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148.

Note that, to prove that an instance of Replacement holds in the model, 𝜑 would need have all instances of ∈ replaced with 𝑅. But this still results in an instance of this theorem, so we do establish that Replacement holds. (Contributed by Eric Schmidt, 6-Nov-2025.)

𝐹:V–1-1-onto→V    &   𝑅 = (◡𝐹 ∘ E )    ⇒   (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧𝑅𝑦 ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)))
 
6-Nov-2025permaxext 45122 The Axiom of Extensionality ax-ext 2705 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
𝐹:V–1-1-onto→V    &   𝑅 = (◡𝐹 ∘ E )    ⇒   (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)
 
6-Nov-2025brpermmodelcnv 45121 Ordinary membership expressed in terms of the permutation model's membership relation. (Contributed by Eric Schmidt, 6-Nov-2025.)
𝐹:V–1-1-onto→V    &   𝑅 = (◡𝐹 ∘ E )    &   𝐴 ∈ V    &   𝐵 ∈ V    ⇒   (𝐴𝑅(◡𝐹‘𝐵) ↔ 𝐴 ∈ 𝐵)
 
6-Nov-2025brpermmodel 45120 The membership relation in a permutation model. We use a permutation 𝐹 of the universe to define a relation 𝑅 that serves as the membership relation in our model. The conclusion of this theorem is Definition II.9.1 of [Kunen2] p. 148. All the axioms of ZFC except for Regularity hold in permutation models, and Regularity will be false if 𝐹 is chosen appropriately. Thus, permutation models can be used to show that Regularity does not follow from the other axioms (with the usual proviso that the axioms are consistent). (Contributed by Eric Schmidt, 6-Nov-2025.)
𝐹:V–1-1-onto→V    &   𝑅 = (◡𝐹 ∘ E )    &   𝐴 ∈ V    &   𝐵 ∈ V    ⇒   (𝐴𝑅𝐵 ↔ 𝐴 ∈ (𝐹‘𝐵))
 
6-Nov-2025orbitclmpt 45075 Version of orbitcl 45074 using maps-to notation. (Contributed by Eric Schmidt, 6-Nov-2025.)
Ⅎ𝑥𝐵    &   Ⅎ𝑥𝐷    &   𝑍 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) “ ω)    &   (𝑥 = 𝐵 → 𝐶 = 𝐷)    ⇒   ((𝐵 ∈ 𝑍 ∧ 𝐷 ∈ 𝑉) → 𝐷 ∈ 𝑍)
 
6-Nov-2025orbitcl 45074 The orbit under a function is closed under the function. (Contributed by Eric Schmidt, 6-Nov-2025.)
(𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ (rec(𝐹, 𝐴) “ ω))
 
6-Nov-2025orbitinit 45073 A set is contained in its orbit. (Contributed by Eric Schmidt, 6-Nov-2025.)
(𝐴 ∈ 𝑉 → 𝐴 ∈ (rec(𝐹, 𝐴) “ ω))
 
6-Nov-2025orbitex 45072 Orbits exist. Given a set 𝐴 and a function 𝐹, the orbit of 𝐴 under 𝐹 is the smallest set 𝑍 such that 𝐴 ∈ 𝑍 and 𝑍 is closed under 𝐹. (Contributed by Eric Schmidt, 6-Nov-2025.)
(rec(𝐹, 𝐴) “ ω) ∈ V
 
6-Nov-2025constrsqrtcl 33813 Constructible numbers are closed under taking the square root. This is not generally the case for the cubic root operation, see 2sqr3nconstr 33815. Item (5) of Theorem 7.10 of [Stewart] p. 96 (Proposed by Saveliy Skresanov, 3-Nov-2025.) (Contributed by Thierry Arnoux, 6-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    ⇒   (𝜑 → (√‘𝑋) ∈ Constr)
 
6-Nov-2025constrabscl 33812 Constructible numbers are closed under absolute value (modulus). (Contributed by Thierry Arnoux, 6-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    ⇒   (𝜑 → (abs‘𝑋) ∈ Constr)
 
6-Nov-2025onsis 28209 Transfinite induction schema for surreal ordinals. (Contributed by Scott Fenton, 6-Nov-2025.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   (𝑥 ∈ Ons → (∀𝑦 ∈ Ons (𝑦 <s 𝑥 → 𝜓) → 𝜑))    ⇒   (𝐴 ∈ Ons → 𝜒)
 
6-Nov-2025onsse 28208 Surreal less-than is set-like over the surreal ordinals. (Contributed by Scott Fenton, 6-Nov-2025.)
<s Se Ons
 
6-Nov-2025onswe 28207 Surreal less-than well-orders the surreal ordinals. Part of Theorem 15 of [Conway] p. 28. (Contributed by Scott Fenton, 6-Nov-2025.)
<s We Ons
 
6-Nov-2025bday11on 28203 The birthday function is one-to-one over the surreal ordinals. (Contributed by Scott Fenton, 6-Nov-2025.)
((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 = 𝐵)
 
6-Nov-2025onsleft 28198 The left set of a surreal ordinal is the same as its old set. (Contributed by Scott Fenton, 6-Nov-2025.)
(𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
 
5-Nov-2025ranup 49767 The universal property of the right Kan extension; expressed explicitly. (Contributed by Zhi Wang, 5-Nov-2025.)
𝑆 = (𝐶 FuncCat 𝐸)    &   𝑀 = (𝐷 Nat 𝐸)    &   𝑁 = (𝐶 Nat 𝐸)    &    ∙ = (comp‘𝑆)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐴 ∈ ((𝐿 ∘func 𝐹)𝑁𝑋))    ⇒   (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴 ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)∃!𝑏 ∈ (𝑙𝑀𝐿)𝑎 = (𝐴(⟨(𝑙 ∘func 𝐹), (𝐿 ∘func 𝐹)⟩ ∙ 𝑋)(𝑏 ∘ (1st ‘𝐹)))))
 
5-Nov-2025incat 49726 Constructing a category with at most one object and at most two morphisms. If 𝑋 is a set then 𝐶 is the category 𝐴 in Exercise 3G of [Adamek] p. 45. (Contributed by Zhi Wang, 5-Nov-2025.)
𝐶 = {⟨(Base‘ndx), {𝑋}⟩, ⟨(Hom ‘ndx), {⟨𝑋, 𝑋, 𝐻⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑋, 𝑋⟩, 𝑋, · ⟩}⟩}    &   𝐻 = {𝐹, 𝐺}    &    · = (𝑓 ∈ 𝐻, 𝑔 ∈ 𝐻 ↦ (𝑓 ∩ 𝑔))    ⇒   ((𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {𝑋} ↦ 𝐺)))
 
5-Nov-20252arwcat 49725 The condition for a structure with at most one object and at most two morphisms being a category. "2arwcat.2" to "2arwcat.5" are also necessary conditions if 𝑋, 0, and 1 are all sets, due to catlid 17591, catrid 17592, and catcocl 17593. (Contributed by Zhi Wang, 5-Nov-2025.)
(𝜑 → {𝑋} = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → · = (comp‘𝐶))    &   (𝑋𝐻𝑋) = { 0 , 1 }    &   (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 1 )    &   (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = 0 )    &   (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 0 )    &   (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) ∈ { 0 , 1 })    ⇒   (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {𝑋} ↦ 1 )))
 
5-Nov-20252arwcatlem5 49724 Lemma for 2arwcat 49725. (Contributed by Zhi Wang, 5-Nov-2025.)
(𝜑 → ( 1 · 0 ) = 0 )    &   (𝜑 → ( 0 · 1 ) = 0 )    &   (𝜑 → ( 0 · 0 ) ∈ { 0 , 1 })    ⇒   (𝜑 → (( 0 · 0 ) · 0 ) = ( 0 · ( 0 · 0 )))
 
5-Nov-20252arwcatlem4 49723 Lemma for 2arwcat 49725. (Contributed by Zhi Wang, 5-Nov-2025.)
(𝜑 → 𝐴 = 𝑋)    &   (𝜑 → 𝐵 = 𝑌)    &   (𝜑 → 𝐶 = 𝑍)    &   (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))    &   (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )    &   (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )    &   (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )    &   (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) ∈ { 0 , 1 })    &   (𝜑 → (𝐺 = 0 ∨ 𝐺 = 1 ))    ⇒   (𝜑 → (𝐺(⟨𝐴, 𝐵⟩ · 𝐶)𝐹) ∈ { 0 , 1 })
 
5-Nov-20252arwcatlem3 49722 Lemma for 2arwcat 49725. (Contributed by Zhi Wang, 5-Nov-2025.)
(𝜑 → 𝐴 = 𝑋)    &   (𝜑 → 𝐵 = 𝑌)    &   (𝜑 → 𝐶 = 𝑍)    &   (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))    &   (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )    &   (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )    ⇒   (𝜑 → (𝐹(⟨𝐴, 𝐵⟩ · 𝐶) 1 ) = 𝐹)
 
5-Nov-20252arwcatlem2 49721 Lemma for 2arwcat 49725. (Contributed by Zhi Wang, 5-Nov-2025.)
(𝜑 → 𝐴 = 𝑋)    &   (𝜑 → 𝐵 = 𝑌)    &   (𝜑 → 𝐶 = 𝑍)    &   (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))    &   (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )    &   (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )    ⇒   (𝜑 → ( 1 (⟨𝐴, 𝐵⟩ · 𝐶)𝐹) = 𝐹)
 
5-Nov-20252arwcatlem1 49720 Lemma for 2arwcat 49725. (Contributed by Zhi Wang, 5-Nov-2025.)
(𝑋𝐻𝑋) = { 0 , 1 }    ⇒   ((((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 ))) ↔ ((𝑥 ∈ {𝑋} ∧ 𝑦 ∈ {𝑋}) ∧ (𝑧 ∈ {𝑋} ∧ 𝑤 ∈ {𝑋}) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
 
5-Nov-2025catcofval 49353 Composition of the category structure. (Contributed by Zhi Wang, 5-Nov-2025.)
𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩}    &    · ∈ V    ⇒    · = (comp‘𝐶)
 
5-Nov-2025cathomfval 49352 The hom-sets of the category structure. (Contributed by Zhi Wang, 5-Nov-2025.)
𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩}    &   𝐻 ∈ V    ⇒   𝐻 = (Hom ‘𝐶)
 
5-Nov-2025catbas 49351 The base of the category structure. (Contributed by Zhi Wang, 5-Nov-2025.)
𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩}    &   𝐵 ∈ V    ⇒   𝐵 = (Base‘𝐶)
 
5-Nov-2025upciclem1 49291 Lemma for upcic 49295, upeu 49296, and upeu2 49297. (Contributed by Zhi Wang, 16-Sep-2025.) (Proof shortened by Zhi Wang, 5-Nov-2025.)
(𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑛 ∈ (𝑍𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑛 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀))    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)))    ⇒   (𝜑 → ∃!𝑙 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑙)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
 
5-Nov-2025nelsubc3 49196 Remark 4.2(2) of [Adamek] p. 48. There exists a set satisfying all conditions for a subcategory but the existence of identity morphisms. Therefore such condition in df-subc 17721 is necessary.

Note that this theorem cheated a little bit because (𝐶 ↾cat 𝐽) is not a category. In fact (𝐶 ↾cat 𝐽) ∈ Cat is a stronger statement than the condition (d) of Definition 4.1(1) of [Adamek] p. 48, as stated here (see the proof of issubc3 17758). To construct such a category, see setc1onsubc 49727 and cnelsubc 49729. (Contributed by Zhi Wang, 5-Nov-2025.)

∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ (¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))))
 
5-Nov-2025nelsubc3lem 49195 Lemma for nelsubc3 49196. (Contributed by Zhi Wang, 5-Nov-2025.)
𝐶 ∈ Cat    &   𝐽 ∈ V    &   𝑆 ∈ V    &   (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat (Homf ‘𝐶) ∧ (¬ ∀𝑥 ∈ 𝑆 ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))))    ⇒   ∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ (¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))))
 
5-Nov-2025nelsubc2 49194 An empty "hom-set" for non-empty base is not a subcategory. (Contributed by Zhi Wang, 5-Nov-2025.)
𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑆 ⊆ 𝐵)    &   (𝜑 → 𝑆 ≠ ∅)    &   (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅}))    &   (𝜑 → 𝐶 ∈ Cat)    ⇒   (𝜑 → ¬ 𝐽 ∈ (Subcat‘𝐶))
 
5-Nov-2025nelsubc 49193 An empty "hom-set" for non-empty base satisfies all conditions for a subcategory but the existence of identity morphisms. (Contributed by Zhi Wang, 5-Nov-2025.)
𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑆 ⊆ 𝐵)    &   (𝜑 → 𝑆 ≠ ∅)    &   (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅}))    &   𝐻 = (Homf ‘𝐶)    &    1 = (Id‘𝐶)    &    · = (comp‘𝐶)    ⇒   (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
 
5-Nov-2025nelsubclem 49192 Lemma for nelsubc 49193. (Contributed by Zhi Wang, 5-Nov-2025.)
𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑆 ⊆ 𝐵)    &   (𝜑 → 𝑆 ≠ ∅)    &   (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅}))    &   𝐻 = (Homf ‘𝐶)    ⇒   (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓))))
 
5-Nov-2025gpgprismgr4cyclex 48231 The generalized Petersen graphs G(N,1), which are the N-prisms, have (at least) one cycle of length 4. (Contributed by AV, 5-Nov-2025.)
(𝑁 ∈ (ℤ≥‘3) → ∃𝑝∃𝑓(𝑓(Cycles‘(𝑁 gPetersenGr 1))𝑝 ∧ (♯‘𝑓) = 4))
 
5-Nov-2025gpgprismgr4cycl0 48230 The generalized Petersen graphs G(N,1), which are the N-prisms, have a cycle of length 4 starting at the vertex ⟨0, 0⟩. (Contributed by AV, 5-Nov-2025.)
𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    &   𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    &   𝐺 = (𝑁 gPetersenGr 1)    ⇒   (𝑁 ∈ (ℤ≥‘3) → (𝐹(Cycles‘𝐺)𝑃 ∧ (♯‘𝐹) = 4))
 
5-Nov-2025gpgprismgr4cycllem11 48229 Lemma 11 for gpgprismgr4cycl0 48230. (Contributed by AV, 5-Nov-2025.)
𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    &   𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    &   𝐺 = (𝑁 gPetersenGr 1)    ⇒   (𝑁 ∈ (ℤ≥‘3) → 𝐹(Cycles‘𝐺)𝑃)
 
5-Nov-2025gpgprismgr4cycllem10 48228 Lemma 10 for gpgprismgr4cycl0 48230. (Contributed by AV, 5-Nov-2025.)
𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    &   𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    &   𝐺 = (𝑁 gPetersenGr 1)    ⇒   ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑋)) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
 
5-Nov-2025gpgprismgr4cycllem3 48221 Lemma 3 for gpgprismgr4cycl0 48230. (Contributed by AV, 5-Nov-2025.)
𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    ⇒   ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑋 ∈ (0..^4)) → ((𝐹‘𝑋) ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∧ ∃𝑥 ∈ (0..^𝑁)((𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ (𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ (𝐹‘𝑋) = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
 
5-Nov-2025constrresqrtcl 33811 If a positive real number 𝑋 is constructible, then, so is its square root. (Contributed by Thierry Arnoux, 5-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    &   (𝜑 → 𝑋 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝑋)    ⇒   (𝜑 → (√‘𝑋) ∈ Constr)
 
5-Nov-2025constrfld 33810 The constructible numbers form a field. (Contributed by Thierry Arnoux, 5-Nov-2025.)
(ℂfld ↾s Constr) ∈ Field
 
5-Nov-2025constrsdrg 33809 Constructible numbers form a subfield of the complex numbers. (Contributed by Thierry Arnoux, 5-Nov-2025.)
Constr ∈ (SubDRing‘ℂfld)
 
5-Nov-2025constrinvcl 33807 Constructible numbers are closed under complex inverse. Item (4) of Theorem 7.10 of [Stewart] p. 96 (Contributed by Thierry Arnoux, 5-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    &   (𝜑 → 𝑋 ≠ 0)    ⇒   (𝜑 → (1 / 𝑋) ∈ Constr)
 
5-Nov-2025constrreinvcl 33806 If a real number 𝑋 is constructible, then, so is its inverse. (Contributed by Thierry Arnoux, 5-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    &   (𝜑 → 𝑋 ≠ 0)    &   (𝜑 → 𝑋 ∈ ℝ)    ⇒   (𝜑 → (1 / 𝑋) ∈ Constr)
 
5-Nov-2025constrmulcl 33805 Constructible numbers are closed under complex multiplication. Item (3) of Theorem 7.10 of [Stewart] p. 96 (Contributed by Thierry Arnoux, 5-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    &   (𝜑 → 𝑌 ∈ Constr)    ⇒   (𝜑 → (𝑋 · 𝑌) ∈ Constr)
 
5-Nov-2025constrimcl 33804 Constructible numbers are closed under taking the imaginary part. (Contributed by Thierry Arnoux, 5-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    ⇒   (𝜑 → (ℑ‘𝑋) ∈ Constr)
 
5-Nov-2025constrrecl 33803 Constructible numbers are closed under taking the real part. (Contributed by Thierry Arnoux, 5-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    ⇒   (𝜑 → (ℜ‘𝑋) ∈ Constr)
 
5-Nov-2025constrcjcl 33802 Constructible numbers are closed under complex conjugate. (Contributed by Thierry Arnoux, 5-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    ⇒   (𝜑 → (∗‘𝑋) ∈ Constr)
 
5-Nov-2025argcj 32736 The argument of the conjugate of a complex number 𝐴. (Contributed by Thierry Arnoux, 5-Nov-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 ≠ 0)    &   (𝜑 → ¬ -𝐴 ∈ ℝ+)    ⇒   (𝜑 → (ℑ‘(log‘(∗‘𝐴))) = -(ℑ‘(log‘𝐴)))
 
5-Nov-2025arginv 32735 The argument of the inverse of a complex number 𝐴. (Contributed by Thierry Arnoux, 5-Nov-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 ≠ 0)    &   (𝜑 → ¬ -𝐴 ∈ ℝ+)    ⇒   (𝜑 → (ℑ‘(log‘(1 / 𝐴))) = -(ℑ‘(log‘𝐴)))
 
5-Nov-2025efiargd 32734 The exponential of the "arg" function ℑ ∘ log, deduction version. (Contributed by Thierry Arnoux, 5-Nov-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 ≠ 0)    ⇒   (𝜑 → (exp‘(i · (ℑ‘(log‘𝐴)))) = (𝐴 / (abs‘𝐴)))
 
5-Nov-2025binom2subadd 32729 The difference of the squares of the sum and difference of two complex numbers 𝐴 and 𝐵. (Contributed by Thierry Arnoux, 5-Nov-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    ⇒   (𝜑 → (((𝐴 + 𝐵)↑2) − ((𝐴 − 𝐵)↑2)) = (4 · (𝐴 · 𝐵)))
 
5-Nov-2025n0sfincut 28283 The simplest number greater than a finite set of non-negative surreal integers is a non-negative surreal integer. (Contributed by Scott Fenton, 5-Nov-2025.)
((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s)
 
4-Nov-2025lanup 49766 The universal property of the left Kan extension; expressed explicitly. (Contributed by Zhi Wang, 4-Nov-2025.)
𝑆 = (𝐶 FuncCat 𝐸)    &   𝑀 = (𝐷 Nat 𝐸)    &   𝑁 = (𝐶 Nat 𝐸)    &    ∙ = (comp‘𝑆)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)))    ⇒   (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴)))
 
4-Nov-2025ranrcl5 49765 The second component of a right Kan extension is a natural transformation. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    &   𝑁 = (𝐶 Nat 𝐸)    ⇒   (𝜑 → 𝐴 ∈ ((𝐿 ∘func 𝐹)𝑁𝑋))
 
4-Nov-2025ranrcl4 49764 The first component of a right Kan extension is a functor. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
 
4-Nov-2025ranrcl4lem 49763 Lemma for ranrcl4 49764 and ranrcl5 49765. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
 
4-Nov-2025ranrcl3 49762 Reverse closure for right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
 
4-Nov-2025ranrcl2 49761 Reverse closure for right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
 
4-Nov-2025lanrcl5 49760 The second component of a left Kan extension is a natural transformation. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴)    &   𝑁 = (𝐶 Nat 𝐸)    ⇒   (𝜑 → 𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)))
 
4-Nov-2025lanrcl4 49759 The first component of a left Kan extension is a functor. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴)    ⇒   (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
 
4-Nov-2025lanrcl3 49758 Reverse closure for left Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴)    ⇒   (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
 
4-Nov-2025lanrcl2 49757 Reverse closure for left Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴)    ⇒   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
 
4-Nov-2025ranval2 49755 The set of right Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by oppcup2 49333 and oppcup3lem 49331. (Contributed by Zhi Wang, 4-Nov-2025.)
𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))    &   𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))    &   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋))
 
4-Nov-2025isran2 49754 A right Kan extension is a universal pair. (Contributed by Zhi Wang, 4-Nov-2025.)
𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))    &   𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))    &   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)    &   (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   (𝜑 → 𝐿(⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋)𝐴)
 
4-Nov-2025isran 49753 A right Kan extension is a universal pair. (Contributed by Zhi Wang, 4-Nov-2025.)
𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))    &   𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))    &   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)    &   (𝜑 → 𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋))    ⇒   (𝜑 → 𝐿 ∈ (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋))
 
4-Nov-2025islan2 49751 A left Kan extension is a universal pair. (Contributed by Zhi Wang, 4-Nov-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   𝑆 = (𝐶 FuncCat 𝐸)    &   𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)    ⇒   (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 → 𝐿(𝐾(𝑅 UP 𝑆)𝑋)𝐴)
 
4-Nov-2025relran 49749 The set of right Kan extensions is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)
Rel (𝐹(𝑃 Ran 𝐸)𝑋)
 
4-Nov-2025ranrcl 49747 Reverse closure for right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
 
4-Nov-2025ranval 49745 Value of the set of right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   𝑆 = (𝐶 FuncCat 𝐸)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))    &   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)    &   𝑂 = (oppCat‘𝑅)    &   𝑃 = (oppCat‘𝑆)    ⇒   (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋))
 
4-Nov-2025reldmran2 49743 The domain of (𝑃 Ran 𝐸) is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)
Rel dom (𝑃 Ran 𝐸)
 
4-Nov-2025ranfval 49739 Value of the function generating the set of right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   𝑆 = (𝐶 FuncCat 𝐸)    &   (𝜑 → 𝐶 ∈ 𝑈)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐸 ∈ 𝑊)    &   𝑂 = (oppCat‘𝑅)    &   𝑃 = (oppCat‘𝑆)    ⇒   (𝜑 → (⟨𝐶, 𝐷⟩ Ran 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂 UP 𝑃)𝑥)))
 
4-Nov-2025reldmran 49737 The domain of Ran is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)
Rel dom Ran
 
4-Nov-2025ranfn 49735 Ran is a function on ((V × V) × V). (Contributed by Zhi Wang, 4-Nov-2025.)
Ran Fn ((V × V) × V)
 
4-Nov-2025df-ran 49733 Definition of the (local) right Kan extension. Given a functor 𝐹:𝐶⟶𝐷 and a functor 𝑋:𝐶⟶𝐸, the set (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) consists of right Kan extensions of 𝑋 along 𝐹, which are universal pairs from the pre-composition functor given by 𝐹 to 𝑋 (ranval2 49755). The definition in § 3 of Chapter X in p. 236 of Mac Lane, Saunders, Categories for the Working Mathematician, 2nd Edition, Springer Science+Business Media, New York, (1998) [QA169.M33 1998]; available at https://math.mit.edu/~hrm/palestine/maclane-categories.pdf 49755 (retrieved 3 Nov 2025).

A right Kan extension is in the form of ⟨𝐿, 𝐴⟩ where the first component is a functor 𝐿:𝐷⟶𝐸 (ranrcl4 49764) and the second component is a natural transformation 𝐴:𝐿𝐹⟶𝑋 (ranrcl5 49765) where 𝐿𝐹 is the composed functor. Intuitively, the first component 𝐿 can be regarded as the result of an "inverse" of pre-composition; the source category of 𝑋:𝐶⟶𝐸 is "extended" along 𝐹:𝐶⟶𝐷.

The right Kan extension is a generalization of many categorical concepts such as limit. In § 7 of Chapter X of Categories for the Working Mathematician, it is concluded that "the notion of Kan extensions subsumes all the other fundamental concepts of category theory".

This definition was chosen over the other version in the commented out section due to its better reverse closure property.

See df-lan 49732 for the dual concept.

(Contributed by Zhi Wang, 4-Nov-2025.)

Ran = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))
 
4-Nov-2025prcoffunca2 49512 The pre-composition functor is a functor. (Contributed by Zhi Wang, 4-Nov-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   (𝜑 → 𝐸 ∈ Cat)    &   𝑆 = (𝐶 FuncCat 𝐸)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐾, 𝐿⟩)    ⇒   (𝜑 → 𝐾(𝑅 Func 𝑆)𝐿)
 
4-Nov-2025prcofelvv 49505 The pre-composition functor is an ordered pair. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝐹 ∈ 𝑈)    &   (𝜑 → 𝑃 ∈ 𝑉)    ⇒   (𝜑 → (𝑃 −∘F 𝐹) ∈ (V × V))
 
4-Nov-2025oppcup3 49334 The universal property for the universal pair ⟨𝑋, 𝑀⟩ from a functor to an object, expressed explicitly. (Contributed by Zhi Wang, 4-Nov-2025.)
𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &    ∙ = (comp‘𝐸)    &   𝑂 = (oppCat‘𝐷)    &   𝑃 = (oppCat‘𝐸)    &   (𝜑 → 𝑋(⟨𝐹, 𝑇⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   (𝜑 → tpos 𝑇 = 𝐺)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑊))    ⇒   (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑌𝐺𝑋)‘𝑘)))
 
4-Nov-2025oppcup2 49333 The universal property for the universal pair ⟨𝑋, 𝑀⟩ from a functor to an object, expressed explicitly. (Contributed by Zhi Wang, 4-Nov-2025.)
𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &    ∙ = (comp‘𝐸)    &   𝑂 = (oppCat‘𝐷)    &   𝑃 = (oppCat‘𝐸)    &   (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑 → 𝑋(⟨𝐹, tpos 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    ⇒   (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘)))
 
4-Nov-2025oppcup3lem 49331 Lemma for oppcup3 49334. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑛 ∈ ((𝐹‘𝑦)𝐽𝑍)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑛 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩𝑂𝑍)((𝑦𝐺𝑋)‘𝑘)))    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑍))    ⇒   (𝜑 → ∃!𝑙 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑙)))
 
4-Nov-2025oppcuprcl2 49327 Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   𝑃 = (oppCat‘𝐸)    &   𝑂 = (oppCat‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑈)    &   (𝜑 → 𝐸 ∈ 𝑉)    &   (𝜑 → tpos 𝐺 = 𝐻)    ⇒   (𝜑 → 𝐹(𝐷 Func 𝐸)𝐻)
 
4-Nov-2025oppcuprcl5 49326 Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   𝑃 = (oppCat‘𝐸)    &   𝐽 = (Hom ‘𝐸)    ⇒   (𝜑 → 𝑀 ∈ ((𝐹‘𝑋)𝐽𝑊))
 
4-Nov-2025oppcuprcl3 49325 Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   𝑃 = (oppCat‘𝐸)    &   𝐶 = (Base‘𝐸)    ⇒   (𝜑 → 𝑊 ∈ 𝐶)
 
4-Nov-2025oppcuprcl4 49324 Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   𝑂 = (oppCat‘𝐷)    &   𝐵 = (Base‘𝐷)    ⇒   (𝜑 → 𝑋 ∈ 𝐵)
 
4-Nov-2025uptpos 49323 Rewrite the predicate of universal property in the form of opposite functor. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   (𝜑 → tpos 𝐺 = 𝐻)    ⇒   (𝜑 → 𝑋(⟨𝐹, tpos 𝐻⟩(𝑂 UP 𝑃)𝑊)𝑀)
 
4-Nov-2025uptposlem 49322 Lemma for uptpos 49323. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   (𝜑 → tpos 𝐺 = 𝐻)    ⇒   (𝜑 → tpos 𝐻 = 𝐺)
 
4-Nov-2025funcoppc3 49272 A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 4-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    &   (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)    &   (𝜑 → 𝐺 Fn (𝐴 × 𝐵))    ⇒   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
 
4-Nov-2025funcoppc2 49268 A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 4-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    &   (𝜑 → 𝐹(𝑂 Func 𝑃)𝐺)    ⇒   (𝜑 → 𝐹(𝐶 Func 𝐷)tpos 𝐺)
 
4-Nov-2025oppfoppc 49266 The opposite functor is a functor on opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   (𝜑 → (𝐹 oppFunc 𝐺) ∈ (𝑂 Func 𝑃))
 
4-Nov-2025oppfval 49261 Value of the opposite functor. (Contributed by Zhi Wang, 4-Nov-2025.)
(𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 oppFunc 𝐺) = ⟨𝐹, tpos 𝐺⟩)
 
4-Nov-2025onsfi 28284 A surreal ordinal with a finite birthday is a non-negative surreal integer. (Contributed by Scott Fenton, 4-Nov-2025.)
((𝐴 ∈ Ons ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℕ0s)
 
4-Nov-2025onscutlt 28202 A surreal ordinal is the simplest number greater than all previous surreal ordinals. Theorem 15 of [Conway] p. 28. (Contributed by Scott Fenton, 4-Nov-2025.)
(𝐴 ∈ Ons → 𝐴 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} |s ∅))
 
3-Nov-2025lanval2 49752 The set of left Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by isup2 49319 and upciclem1 49291. (Contributed by Zhi Wang, 3-Nov-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   𝑆 = (𝐶 FuncCat 𝐸)    &   𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)    ⇒   (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))
 
3-Nov-2025islan 49750 A left Kan extension is a universal pair. (Contributed by Zhi Wang, 3-Nov-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   𝑆 = (𝐶 FuncCat 𝐸)    &   𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)    ⇒   (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝐿 ∈ (𝐾(𝑅 UP 𝑆)𝑋))
 
3-Nov-2025rellan 49748 The set of left Kan extensions is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)
Rel (𝐹(𝑃 Lan 𝐸)𝑋)
 
3-Nov-2025lanrcl 49746 Reverse closure for left Kan extensions. (Contributed by Zhi Wang, 3-Nov-2025.)
(𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
 
3-Nov-2025lanval 49744 Value of the set of left Kan extensions. (Contributed by Zhi Wang, 3-Nov-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   𝑆 = (𝐶 FuncCat 𝐸)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))    &   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = 𝐾)    ⇒   (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))
 
3-Nov-2025reldmlan2 49742 The domain of (𝑃 Lan 𝐸) is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)
Rel dom (𝑃 Lan 𝐸)
 
3-Nov-2025lanfval 49738 Value of the function generating the set of left Kan extensions. (Contributed by Zhi Wang, 3-Nov-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   𝑆 = (𝐶 FuncCat 𝐸)    &   (𝜑 → 𝐶 ∈ 𝑈)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐸 ∈ 𝑊)    ⇒   (𝜑 → (⟨𝐶, 𝐷⟩ Lan 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)))
 
3-Nov-2025reldmlan 49736 The domain of Lan is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)
Rel dom Lan
 
3-Nov-2025lanfn 49734 Lan is a function on ((V × V) × V). (Contributed by Zhi Wang, 3-Nov-2025.)
Lan Fn ((V × V) × V)
 
3-Nov-2025df-lan 49732 Definition of the (local) left Kan extension. Given a functor 𝐹:𝐶⟶𝐷 and a functor 𝑋:𝐶⟶𝐸, the set (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) consists of left Kan extensions of 𝑋 along 𝐹, which are universal pairs from 𝑋 to the pre-composition functor given by 𝐹 (lanval2 49752). See also § 3 of Chapter X in p. 240 of Mac Lane, Saunders, Categories for the Working Mathematician, 2nd Edition, Springer Science+Business Media, New York, (1998) [QA169.M33 1998]; available at https://math.mit.edu/~hrm/palestine/maclane-categories.pdf 49752 (retrieved 3 Nov 2025).

A left Kan extension is in the form of ⟨𝐿, 𝐴⟩ where the first component is a functor 𝐿:𝐷⟶𝐸 (lanrcl4 49759) and the second component is a natural transformation 𝐴:𝑋⟶𝐿𝐹 (lanrcl5 49760) where 𝐿𝐹 is the composed functor. Intuitively, the first component 𝐿 can be regarded as the result of an "inverse" of pre-composition; the source category of 𝑋:𝐶⟶𝐸 is "extended" along 𝐹:𝐶⟶𝐷.

The left Kan extension is a generalization of many categorical concepts such as colimit. In § 7 of Chapter X of Categories for the Working Mathematician, it is concluded that "the notion of Kan extensions subsumes all the other fundamental concepts of category theory".

This definition was chosen over the other version in the commented out section due to its better reverse closure property.

See df-ran 49733 for the dual concept.

(Contributed by Zhi Wang, 3-Nov-2025.)

Lan = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)))
 
3-Nov-2025prcof22a 49517 The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)
𝑁 = (𝐷 Nat 𝐸)    &   (𝜑 → 𝐴 ∈ (𝐾𝑁𝐿))    &   (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = (𝐴‘((1st ‘𝐹)‘𝑋)))
 
3-Nov-2025prcof21a 49516 The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)
𝑁 = (𝐷 Nat 𝐸)    &   (𝜑 → 𝐴 ∈ (𝐾𝑁𝐿))    &   (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑈)    ⇒   (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))
 
3-Nov-2025prcof2 49515 The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)
𝑁 = (𝐷 Nat 𝐸)    &   (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))    &   (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩)) = 𝑃)    &   Rel 𝑅    &   (𝜑 → 𝐹𝑅𝐺)    ⇒   (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ 𝐹)))
 
3-Nov-2025prcof2a 49514 The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)
𝑁 = (𝐷 Nat 𝐸)    &   (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))    &   (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑈)    ⇒   (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))))
 
3-Nov-2025prcof1 49513 The object part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)
(𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))    &   (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑂)    ⇒   (𝜑 → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))
 
3-Nov-2025cicerALT 49171 Isomorphism is an equivalence relation on objects of a category. Remark 3.16 in [Adamek] p. 29. (Contributed by AV, 5-Apr-2020.) (Proof shortened by Zhi Wang, 3-Nov-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) Er (Base‘𝐶))
 
3-Nov-2025isofnALT 49156 The function value of the function returning the isomorphisms of a category is a function over the Cartesian square of the base set of the category. (Contributed by AV, 5-Apr-2020.) (Proof shortened by Zhi Wang, 3-Nov-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
 
3-Nov-2025gpgprismgr4cycllem9 48227 Lemma 9 for gpgprismgr4cycl0 48230. (Contributed by AV, 3-Nov-2025.)
𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    &   𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    &   𝐺 = (𝑁 gPetersenGr 1)    ⇒   (𝑁 ∈ (ℤ≥‘3) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
 
3-Nov-2025zconstr 33798 Integers are constructible. (Contributed by Thierry Arnoux, 3-Nov-2025.)
(𝜑 → 𝑋 ∈ ℤ)    ⇒   (𝜑 → 𝑋 ∈ Constr)
 
2-Nov-2025prcoffunca 49511 The pre-composition functor is a functor. (Contributed by Zhi Wang, 2-Nov-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   (𝜑 → 𝐸 ∈ Cat)    &   𝑆 = (𝐶 FuncCat 𝐸)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (𝑅 Func 𝑆))
 
2-Nov-2025prcoffunc 49510 The pre-composition functor is a functor. (Contributed by Zhi Wang, 2-Nov-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   (𝜑 → 𝐸 ∈ Cat)    &   𝑆 = (𝐶 FuncCat 𝐸)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) ∈ (𝑅 Func 𝑆))
 
2-Nov-2025prcoftposcurfucoa 49509 The pre-composition functor is the transposed curry of the functor composition bifunctor. (Contributed by Zhi Wang, 2-Nov-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   (𝜑 → 𝐸 ∈ Cat)    &   𝑄 = (𝐶 FuncCat 𝐷)    &   (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))    &   (𝜑 → 𝑀 = ((1st ‘ ⚬ )‘𝐹))    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = 𝑀)
 
2-Nov-2025prcoftposcurfuco 49508 The pre-composition functor is the transposed curry of the functor composition bifunctor. (Contributed by Zhi Wang, 2-Nov-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   (𝜑 → 𝐸 ∈ Cat)    &   𝑄 = (𝐶 FuncCat 𝐷)    &   (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))    &   (𝜑 → 𝑀 = ((1st ‘ ⚬ )‘⟨𝐹, 𝐺⟩))    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) = 𝑀)
 
2-Nov-2025reldmprcof2 49507 The domain of the morphism part of the pre-composition functor is a relation. (Contributed by Zhi Wang, 2-Nov-2025.)
Rel dom (2nd ‘(𝑃 −∘F 𝐹))
 
2-Nov-2025reldmprcof1 49506 The domain of the object part of the pre-composition functor is a relation. (Contributed by Zhi Wang, 2-Nov-2025.)
Rel dom (1st ‘(𝑃 −∘F 𝐹))
 
2-Nov-2025prcofval 49503 Value of the pre-composition functor. (Contributed by Zhi Wang, 2-Nov-2025.)
𝐵 = (𝐷 Func 𝐸)    &   𝑁 = (𝐷 Nat 𝐸)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐸 ∈ 𝑊)    &   Rel 𝑅    &   (𝜑 → 𝐹𝑅𝐺)    ⇒   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))⟩)
 
2-Nov-2025prcofvala 49502 Value of the pre-composition functor. (Contributed by Zhi Wang, 2-Nov-2025.)
𝐵 = (𝐷 Func 𝐸)    &   𝑁 = (𝐷 Nat 𝐸)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐸 ∈ 𝑊)    &   (𝜑 → 𝐹 ∈ 𝑈)    ⇒   (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
 
2-Nov-2025prcofvalg 49501 Value of the pre-composition functor. (Contributed by Zhi Wang, 2-Nov-2025.)
𝐵 = (𝐷 Func 𝐸)    &   𝑁 = (𝐷 Nat 𝐸)    &   (𝜑 → 𝐹 ∈ 𝑈)    &   (𝜑 → 𝑃 ∈ 𝑉)    &   (𝜑 → (1st ‘𝑃) = 𝐷)    &   (𝜑 → (2nd ‘𝑃) = 𝐸)    ⇒   (𝜑 → (𝑃 −∘F 𝐹) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
 
2-Nov-2025reldmprcof 49500 The domain of −∘F is a relation. (Contributed by Zhi Wang, 2-Nov-2025.)
Rel dom −∘F
 
2-Nov-2025df-prcof 49499 Definition of pre-composition functors. The object part of the pre-composition functor given by 𝐹 pre-composes a functor with 𝐹; the morphism part pre-composes a natural transformation with the object part of 𝐹, in terms of function composition. Comments before the definition in § 3 of Chapter X in p. 236 of Mac Lane, Saunders, Categories for the Working Mathematician, 2nd Edition, Springer Science+Business Media, New York, (1998) [QA169.M33 1998]; available at https://math.mit.edu/~hrm/palestine/maclane-categories.pdf (retrieved 3 Nov 2025). The notation −∘F is inspired by this page: https://1lab.dev/Cat.Functor.Compose.html.

The pre-composition functor can also be defined as a transposed curry of the functor composition bifunctor (precofval3 49496). But such definition requires an explicit third category. prcoftposcurfuco 49508 and prcoftposcurfucoa 49509 prove the equivalence. (Contributed by Zhi Wang, 2-Nov-2025.)

−∘F = (𝑝 ∈ V, 𝑓 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑑⦌⦋(2nd ‘𝑝) / 𝑒⦌⦋(𝑑 Func 𝑒) / 𝑏⦌⟨(𝑘 ∈ 𝑏 ↦ (𝑘 ∘func 𝑓)), (𝑘 ∈ 𝑏, 𝑙 ∈ 𝑏 ↦ (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1st ‘𝑓))))⟩)
 
2-Nov-2025gpgprismgr4cycllem8 48226 Lemma 8 for gpgprismgr4cycl0 48230. (Contributed by AV, 2-Nov-2025.)
𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    &   𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    &   𝐺 = (𝑁 gPetersenGr 1)    ⇒   (𝑁 ∈ (ℤ≥‘3) → 𝐹 ∈ Word dom (iEdg‘𝐺))
 
2-Nov-2025gpgprismgr4cycllem2 48220 Lemma 2 for gpgprismgr4cycl0 48230: the cycle ⟨𝑃, 𝐹⟩ is proper, i.e., it has no overlapping edges. (Contributed by AV, 2-Nov-2025.)
𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    ⇒   Fun ◡𝐹
 
2-Nov-2025gpgprismgriedgdmss 48176 A subset of the index of edges of the generalized Petersen graph GPG(N,1). (Contributed by AV, 2-Nov-2025.)
(𝑁 ∈ (ℤ≥‘3) → ({{⟨0, 0⟩, ⟨0, 1⟩}, {⟨0, 0⟩, ⟨1, 0⟩}} ∪ {{⟨1, 1⟩, ⟨0, 1⟩}, {⟨1, 1⟩, ⟨1, 0⟩}}) ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))
 
2-Nov-2025gpgprismgriedgdmel 48175 An index of edges of the generalized Petersen graph GPG(N,1). (Contributed by AV, 2-Nov-2025.)
𝐼 = (0..^𝑁)    &   𝐺 = (𝑁 gPetersenGr 1)    ⇒   (𝑁 ∈ (ℤ≥‘3) → (𝑋 ∈ dom (iEdg‘𝐺) ↔ ∃𝑥 ∈ 𝐼 (𝑋 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑋 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑋 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
 
2-Nov-2025gpgiedgdmel 48173 An index of edges of the generalized Petersen graph GPG(N,K). (Contributed by AV, 2-Nov-2025.)
𝐼 = (0..^𝑁)    &   𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   𝐺 = (𝑁 gPetersenGr 𝐾)    ⇒   ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ dom (iEdg‘𝐺) ↔ ∃𝑥 ∈ 𝐼 (𝑋 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑋 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑋 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
 
2-Nov-2025gpgiedgdmellem 48170 Lemma for gpgiedgdmel 48173 and gpgedgel 48174. (Contributed by AV, 2-Nov-2025.)
𝐼 = (0..^𝑁)    &   𝐽 = (1..^(⌈‘(𝑁 / 2)))    ⇒   ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (∃𝑥 ∈ 𝐼 (𝑌 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑌 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑌 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}) → 𝑌 ∈ 𝒫 ({0, 1} × 𝐼)))
 
2-Nov-20251elfzo1ceilhalf1 47461 1 is in the half-open integer range from 1 to the ceiling of half of an integer greater than two is greater than one. (Contributed by AV, 2-Nov-2025.)
(𝑁 ∈ (ℤ≥‘3) → 1 ∈ (1..^(⌈‘(𝑁 / 2))))
 
2-Nov-2025ceilhalfnn 47460 The ceiling of half of a positive integer is a positive integer. (Contributed by AV, 2-Nov-2025.)
(𝑁 ∈ ℕ → (⌈‘(𝑁 / 2)) ∈ ℕ)
 
2-Nov-2025rehalfge1 47459 Half of a real number greater than or equal to two is greater than or equal to one. (Contributed by AV, 2-Nov-2025.)
(𝑋 ∈ (2[,)+∞) → 1 ≤ (𝑋 / 2))
 
2-Nov-2025ceilhalf1 47458 The ceiling of one half is one. (Contributed by AV, 2-Nov-2025.)
(⌈‘(1 / 2)) = 1
 
2-Nov-2025ceilbi 47457 A condition equivalent to ceiling. Analogous to flbi 13722. (Contributed by AV, 2-Nov-2025.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌈‘𝐴) = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 < (𝐴 + 1))))
 
2-Nov-2025ceilhalfgt1 47453 The ceiling of half of an integer greater than two is greater than one. (Contributed by AV, 2-Nov-2025.)
(𝑁 ∈ (ℤ≥‘3) → 1 < (⌈‘(𝑁 / 2)))
 
2-Nov-2025constrremulcl 33801 If two real numbers 𝑋 and 𝑌 are constructible, then, so is their product. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    &   (𝜑 → 𝑌 ∈ Constr)    &   (𝜑 → 𝑋 ∈ ℝ)    &   (𝜑 → 𝑌 ∈ ℝ)    ⇒   (𝜑 → (𝑋 · 𝑌) ∈ Constr)
 
2-Nov-2025iconstr 33800 The imaginary unit i is constructible. (Contributed by Thierry Arnoux, 2-Nov-2025.)
i ∈ Constr
 
2-Nov-2025constrdircl 33799 Constructible numbers are closed under taking the point on the unit circle having the same argument. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    &   (𝜑 → 𝑋 ≠ 0)    ⇒   (𝜑 → (𝑋 / (abs‘𝑋)) ∈ Constr)
 
2-Nov-2025constrnegcl 33797 Constructible numbers are closed under additive inverse. Item (2) of Theorem 7.10 of [Stewart] p. 96 (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    ⇒   (𝜑 → -𝑋 ∈ Constr)
 
2-Nov-2025constraddcl 33796 Constructive numbers are closed under complex addition. Item (1) of Theorem 7.10 of [Stewart] p. 96 (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    &   (𝜑 → 𝑌 ∈ Constr)    ⇒   (𝜑 → (𝑋 + 𝑌) ∈ Constr)
 
2-Nov-2025nn0constr 33795 Nonnegative integers are constructible. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → 𝑁 ∈ Constr)
 
2-Nov-2025constrcn 33794 Constructible numbers are complex numbers. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝑋 ∈ Constr)    ⇒   (𝜑 → 𝑋 ∈ ℂ)
 
2-Nov-2025constrcccl 33792 Constructible numbers are closed under circle-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝐴 ∈ Constr)    &   (𝜑 → 𝐵 ∈ Constr)    &   (𝜑 → 𝐶 ∈ Constr)    &   (𝜑 → 𝐷 ∈ Constr)    &   (𝜑 → 𝐸 ∈ Constr)    &   (𝜑 → 𝐹 ∈ Constr)    &   (𝜑 → 𝑋 ∈ ℂ)    &   (𝜑 → 𝐴 ≠ 𝐷)    &   (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐶)))    &   (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))    ⇒   (𝜑 → 𝑋 ∈ Constr)
 
2-Nov-2025constrlccl 33791 Constructible numbers are closed under line-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝐴 ∈ Constr)    &   (𝜑 → 𝐵 ∈ Constr)    &   (𝜑 → 𝐺 ∈ Constr)    &   (𝜑 → 𝐸 ∈ Constr)    &   (𝜑 → 𝐹 ∈ Constr)    &   (𝜑 → 𝑇 ∈ ℝ)    &   (𝜑 → 𝑋 ∈ ℂ)    &   (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   (𝜑 → (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))    ⇒   (𝜑 → 𝑋 ∈ Constr)
 
2-Nov-2025constrllcl 33790 Constructible numbers are closed under line-line intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝐴 ∈ Constr)    &   (𝜑 → 𝐵 ∈ Constr)    &   (𝜑 → 𝐺 ∈ Constr)    &   (𝜑 → 𝐷 ∈ Constr)    &   (𝜑 → 𝑇 ∈ ℝ)    &   (𝜑 → 𝑅 ∈ ℝ)    &   (𝜑 → 𝑋 ∈ ℂ)    &   (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   (𝜑 → 𝑋 = (𝐺 + (𝑅 · (𝐷 − 𝐺))))    &   (𝜑 → (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐷 − 𝐺))) ≠ 0)    ⇒   (𝜑 → 𝑋 ∈ Constr)
 
2-Nov-2025constrcbvlem 33789 Technical lemma for eliminating the hypothesis of constr0 33771 and co. (Contributed by Thierry Arnoux, 2-Nov-2025.)
rec((𝑧 ∈ V ↦ {𝑦 ∈ ℂ ∣ (∃𝑖 ∈ 𝑧 ∃𝑗 ∈ 𝑧 ∃𝑘 ∈ 𝑧 ∃𝑙 ∈ 𝑧 ∃𝑜 ∈ ℝ ∃𝑝 ∈ ℝ (𝑦 = (𝑖 + (𝑜 · (𝑗 − 𝑖))) ∧ 𝑦 = (𝑘 + (𝑝 · (𝑙 − 𝑘))) ∧ (ℑ‘((∗‘(𝑗 − 𝑖)) · (𝑙 − 𝑘))) ≠ 0) ∨ ∃𝑖 ∈ 𝑧 ∃𝑗 ∈ 𝑧 ∃𝑘 ∈ 𝑧 ∃𝑚 ∈ 𝑧 ∃𝑞 ∈ 𝑧 ∃𝑜 ∈ ℝ (𝑦 = (𝑖 + (𝑜 · (𝑗 − 𝑖))) ∧ (abs‘(𝑦 − 𝑘)) = (abs‘(𝑚 − 𝑞))) ∨ ∃𝑖 ∈ 𝑧 ∃𝑗 ∈ 𝑧 ∃𝑘 ∈ 𝑧 ∃𝑙 ∈ 𝑧 ∃𝑚 ∈ 𝑧 ∃𝑞 ∈ 𝑧 (𝑖 ≠ 𝑙 ∧ (abs‘(𝑦 − 𝑖)) = (abs‘(𝑗 − 𝑘)) ∧ (abs‘(𝑦 − 𝑙)) = (abs‘(𝑚 − 𝑞))))}), {0, 1}) = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})
 
2-Nov-2025constrcccllem 33788 Constructible numbers are closed under circle-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝐴 ∈ Constr)    &   (𝜑 → 𝐵 ∈ Constr)    &   (𝜑 → 𝐺 ∈ Constr)    &   (𝜑 → 𝐷 ∈ Constr)    &   (𝜑 → 𝐸 ∈ Constr)    &   (𝜑 → 𝐹 ∈ Constr)    &   (𝜑 → 𝑋 ∈ ℂ)    &   (𝜑 → 𝐴 ≠ 𝐷)    &   (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)))    &   (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))    ⇒   (𝜑 → 𝑋 ∈ Constr)
 
2-Nov-2025constrlccllem 33787 Constructible numbers are closed under line-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝐴 ∈ Constr)    &   (𝜑 → 𝐵 ∈ Constr)    &   (𝜑 → 𝐺 ∈ Constr)    &   (𝜑 → 𝐸 ∈ Constr)    &   (𝜑 → 𝐹 ∈ Constr)    &   (𝜑 → 𝑇 ∈ ℝ)    &   (𝜑 → 𝑋 ∈ ℂ)    &   (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   (𝜑 → (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))    ⇒   (𝜑 → 𝑋 ∈ Constr)
 
2-Nov-2025constrllcllem 33786 Constructible numbers are closed under line-line intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝐴 ∈ Constr)    &   (𝜑 → 𝐵 ∈ Constr)    &   (𝜑 → 𝐺 ∈ Constr)    &   (𝜑 → 𝐷 ∈ Constr)    &   (𝜑 → 𝑇 ∈ ℝ)    &   (𝜑 → 𝑅 ∈ ℝ)    &   (𝜑 → 𝑋 ∈ ℂ)    &   (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   (𝜑 → 𝑋 = (𝐺 + (𝑅 · (𝐷 − 𝐺))))    &   (𝜑 → (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐷 − 𝐺))) ≠ 0)    ⇒   (𝜑 → 𝑋 ∈ Constr)
 
2-Nov-2025constrfiss 33785 For any finite set 𝐴 of constructible numbers, there is a 𝑛 -th step (𝐶‘𝑛) containing all numbers in 𝐴. (Contributed by Thierry Arnoux, 2-Nov-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝐴 ⊆ Constr)    &   (𝜑 → 𝐴 ∈ Fin)    ⇒   (𝜑 → ∃𝑛 ∈ ω 𝐴 ⊆ (𝐶‘𝑛))
 
2-Nov-2025pythagreim 32733 A simplified version of the Pythagorean theorem, where the points 𝐴 and 𝐵 respectively lie on the imaginary and real axes, and the right angle is at the origin. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    ⇒   (𝜑 → ((abs‘(𝐵 − (i · 𝐴)))↑2) = ((𝐴↑2) + (𝐵↑2)))
 
2-Nov-2025tpsscd 32523 If an ordered triple is a subset of a class, the third element of the triple is an element of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)    ⇒   (𝜑 → 𝐶 ∈ 𝐷)
 
2-Nov-2025tpssbd 32522 If an ordered triple is a subset of a class, the second element of the triple is an element of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)    ⇒   (𝜑 → 𝐵 ∈ 𝐷)
 
2-Nov-2025tpssad 32521 If an ordered triple is a subset of a class, the first element of the triple is an element of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)    ⇒   (𝜑 → 𝐴 ∈ 𝐷)
 
2-Nov-2025tpssd 32520 Deduction version of tpssi : An unordered triple of elements of a class is a subset of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝐴 ∈ 𝐷)    &   (𝜑 → 𝐵 ∈ 𝐷)    &   (𝜑 → 𝐶 ∈ 𝐷)    ⇒   (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
 
2-Nov-2025tpssg 32519 An unordered triple of elements of a class is a subset of the class. (Contributed by Thierry Arnoux, 2-Nov-2025.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷))
 
2-Nov-2025prssbd 32512 If a pair is a subset of a class, the second element of the pair is an element of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)    ⇒   (𝜑 → 𝐵 ∈ 𝐶)
 
2-Nov-2025prssad 32511 If a pair is a subset of a class, the first element of the pair is an element of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)    ⇒   (𝜑 → 𝐴 ∈ 𝐶)
 
2-Nov-20253r19.43 3102 Restricted quantifier version of 19.43 1883 for a triple disjunction . (Contributed by AV, 2-Nov-2025.)
(∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓 ∨ ∃𝑥 ∈ 𝐴 𝜒))
 
1-Nov-2025iinfconstbas 49191 The discrete category is the indexed intersection of all subcategories with the same base. (Contributed by Zhi Wang, 1-Nov-2025.)
𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))    &   𝐵 = (Base‘𝐶)    &   𝐼 = (Id‘𝐶)    &   (𝜑 → 𝑆 ⊆ 𝐵)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)}))    ⇒   (𝜑 → 𝐽 = (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)))
 
1-Nov-2025iinfconstbaslem 49190 Lemma for iinfconstbas 49191. (Contributed by Zhi Wang, 1-Nov-2025.)
𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))    &   𝐵 = (Base‘𝐶)    &   𝐼 = (Id‘𝐶)    &   (𝜑 → 𝑆 ⊆ 𝐵)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)}))    ⇒   (𝜑 → 𝐽 ∈ 𝐴)
 
1-Nov-2025discsubc 49189 A discrete category, whose only morphisms are the identity morphisms, is a subcategory. (Contributed by Zhi Wang, 1-Nov-2025.)
𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))    &   𝐵 = (Base‘𝐶)    &   𝐼 = (Id‘𝐶)    &   (𝜑 → 𝑆 ⊆ 𝐵)    &   (𝜑 → 𝐶 ∈ Cat)    ⇒   (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
 
1-Nov-2025discsubclem 49188 Lemma for discsubc 49189. (Contributed by Zhi Wang, 1-Nov-2025.)
𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))    ⇒   𝐽 Fn (𝑆 × 𝑆)
 
1-Nov-2025dmdm 49178 The double domain of a function on a Cartesian square. (Contributed by Zhi Wang, 1-Nov-2025.)
(𝐴 Fn (𝐵 × 𝐵) → 𝐵 = dom dom 𝐴)
 
1-Nov-2025ixpv 49014 Infinite Cartesian product of the universal class is the set of functions with a fixed domain. (Contributed by Zhi Wang, 1-Nov-2025.)
X𝑥 ∈ 𝐴 V = {𝑓 ∣ 𝑓 Fn 𝐴}
 
1-Nov-2025iinglb 48946 The indexed intersection is the the greatest lower bound if it exists. (Contributed by Zhi Wang, 1-Nov-2025.)
(𝜑 → 𝑋 ∈ 𝐴)    &   ((𝜑 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐶)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ⊆ 𝐵)    ⇒   (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
 
1-Nov-2025iunlub 48945 The indexed union is the the lowest upper bound if it exists. (Contributed by Zhi Wang, 1-Nov-2025.)
(𝜑 → 𝑋 ∈ 𝐴)    &   ((𝜑 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐶)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)    ⇒   (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
 
1-Nov-2025iuneq0 48943 An indexed union is empty iff all indexed classes are empty. (Contributed by Zhi Wang, 1-Nov-2025.)
(∀𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)
 
1-Nov-2025gpgprismgr4cycllem7 48225 Lemma 7 for gpgprismgr4cycl0 48230: the cycle ⟨𝑃, 𝐹⟩ is proper, i.e., it has no overlapping vertices, except the first and the last one. (Contributed by AV, 1-Nov-2025.)
𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    ⇒   ((𝑋 ∈ (0..^(♯‘𝑃)) ∧ 𝑌 ∈ (1..^4)) → (𝑋 ≠ 𝑌 → (𝑃‘𝑋) ≠ (𝑃‘𝑌)))
 
1-Nov-2025gpgprismgr4cycllem6 48224 Lemma 6 for gpgprismgr4cycl0 48230: the cycle ⟨𝑃, 𝐹⟩ is closed, i.e., the first and the last vertex are identical. (Contributed by AV, 1-Nov-2025.)
𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    ⇒   (𝑃‘0) = (𝑃‘4)
 
1-Nov-2025gpgprismgr4cycllem5 48223 Lemma 5 for gpgprismgr4cycl0 48230. (Contributed by AV, 1-Nov-2025.)
𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    ⇒   𝑃 ∈ Word V
 
1-Nov-2025gpgprismgr4cycllem4 48222 Lemma 4 for gpgprismgr4cycl0 48230: the cycle ⟨𝑃, 𝐹⟩ consists of 5 vertices (the first and the last vertex are identical, see gpgprismgr4cycllem6 48224. (Contributed by AV, 1-Nov-2025.)
𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    ⇒   (♯‘𝑃) = 5
 
1-Nov-2025gpgprismgr4cycllem1 48219 Lemma 1 for gpgprismgr4cycl0 48230: the cycle ⟨𝑃, 𝐹⟩ consists of 4 edges (i.e., has length 4). (Contributed by AV, 1-Nov-2025.)
𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    ⇒   (♯‘𝐹) = 4
 
31-Oct-2025infsubc2d 49187 The intersection of two subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.)
(𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   (𝜑 → 𝐽 Fn (𝑇 × 𝑇))    &   (𝜑 → 𝐻 ∈ (Subcat‘𝐶))    &   (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    ⇒   (𝜑 → (𝑥 ∈ (𝑆 ∩ 𝑇), 𝑦 ∈ (𝑆 ∩ 𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))
 
31-Oct-2025infsubc2 49186 The intersection of two subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.)
((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom dom 𝐴 ∩ dom dom 𝐵), 𝑦 ∈ (dom dom 𝐴 ∩ dom dom 𝐵) ↦ ((𝑥𝐴𝑦) ∩ (𝑥𝐵𝑦))) ∈ (Subcat‘𝐶))
 
31-Oct-2025infsubc 49185 The intersection of two subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.)
((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) ∈ (Subcat‘𝐶))
 
31-Oct-2025iinfprg 49184 Indexed intersection of functions with an unordered pair index. (Contributed by Zhi Wang, 31-Oct-2025.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) = (𝑥 ∈ ∩ 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 ↦ ∩ 𝑦 ∈ {𝐴, 𝐵} (𝑦‘𝑥)))
 
31-Oct-2025iinfsubc 49183 Indexed intersection of subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.)
(𝜑 → 𝐴 ≠ ∅)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ∈ (Subcat‘𝐶))    &   (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦)))    ⇒   (𝜑 → 𝐾 ∈ (Subcat‘𝐶))
 
31-Oct-2025iinfssc 49182 Indexed intersection of subcategories is a subcategory (the category-agnostic version). (Contributed by Zhi Wang, 31-Oct-2025.)
(𝜑 → 𝐴 ≠ ∅)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽)    &   (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦)))    ⇒   (𝜑 → 𝐾 ⊆cat 𝐽)
 
31-Oct-2025iinfssclem3 49181 Lemma for iinfssc 49182. (Contributed by Zhi Wang, 31-Oct-2025.)
(𝜑 → 𝐴 ≠ ∅)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽)    &   (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦)))    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = dom dom 𝐻)    &   Ⅎ𝑥𝜑    &   (𝜑 → 𝑋 ∈ ∩ 𝑥 ∈ 𝐴 𝑆)    &   (𝜑 → 𝑌 ∈ ∩ 𝑥 ∈ 𝐴 𝑆)    ⇒   (𝜑 → (𝑋𝐾𝑌) = ∩ 𝑥 ∈ 𝐴 (𝑋𝐻𝑌))
 
31-Oct-2025iinfssclem2 49180 Lemma for iinfssc 49182. (Contributed by Zhi Wang, 31-Oct-2025.)
(𝜑 → 𝐴 ≠ ∅)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽)    &   (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦)))    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = dom dom 𝐻)    &   Ⅎ𝑥𝜑    ⇒   (𝜑 → 𝐾 Fn (∩ 𝑥 ∈ 𝐴 𝑆 × ∩ 𝑥 ∈ 𝐴 𝑆))
 
31-Oct-2025iinfssclem1 49179 Lemma for iinfssc 49182. (Contributed by Zhi Wang, 31-Oct-2025.)
(𝜑 → 𝐴 ≠ ∅)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽)    &   (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦)))    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = dom dom 𝐻)    &   Ⅎ𝑥𝜑    ⇒   (𝜑 → 𝐾 = (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆, 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ↦ ∩ 𝑥 ∈ 𝐴 (𝑧𝐻𝑤)))
 
31-Oct-2025gpgprismgrusgra 48182 The generalized Petersen graphs G(N,1), which are the N-prisms, are simple graphs. (Contributed by AV, 31-Oct-2025.)
(𝑁 ∈ (ℤ≥‘3) → (𝑁 gPetersenGr 1) ∈ USGraph)
 
31-Oct-2025upgrimcycls 48035 Graph isomorphisms between simple pseudographs map cycles onto cycles. (Contributed by AV, 31-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹(Cycles‘𝐺)𝑃)    ⇒   (𝜑 → 𝐸(Cycles‘𝐻)(𝑁 ∘ 𝑃))
 
31-Oct-2025upgrimspths 48034 Graph isomorphisms between simple pseudographs map simple paths onto simple paths. (Contributed by AV, 31-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹(SPaths‘𝐺)𝑃)    ⇒   (𝜑 → 𝐸(SPaths‘𝐻)(𝑁 ∘ 𝑃))
 
31-Oct-2025upgrimpths 48033 Graph isomorphisms between simple pseudographs map paths onto paths. (Contributed by AV, 31-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹(Paths‘𝐺)𝑃)    ⇒   (𝜑 → 𝐸(Paths‘𝐻)(𝑁 ∘ 𝑃))
 
31-Oct-2025upgrimpthslem2 48032 Lemma 2 for upgrimpths 48033. (Contributed by AV, 31-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹(Paths‘𝐺)𝑃)    ⇒   ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘0) ∧ ¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
 
31-Oct-2025squeezedltsq 47010 If a real value is squeezed between two others, its square is less than square of at least one of them. Deduction form. (Contributed by Ender Ting, 31-Oct-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐶 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → 𝐵 < 𝐶)    ⇒   (𝜑 → ((𝐵 · 𝐵) < (𝐴 · 𝐴) ∨ (𝐵 · 𝐵) < (𝐶 · 𝐶)))
 
30-Oct-2025dmrnxp 48961 A Cartesian product is the Cartesian product of its domain and range. (Contributed by Zhi Wang, 30-Oct-2025.)
(𝑅 = (𝐴 × 𝐵) → 𝑅 = (dom 𝑅 × ran 𝑅))
 
30-Oct-2025intxp 48956 Intersection of Cartesian products is the Cartesian product of intersection of domains and ranges. See also inxp 5775 and iinxp 48955. (Contributed by Zhi Wang, 30-Oct-2025.)
(𝜑 → 𝐴 ≠ ∅)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = (dom 𝑥 × ran 𝑥))    &   𝑋 = ∩ 𝑥 ∈ 𝐴 dom 𝑥    &   𝑌 = ∩ 𝑥 ∈ 𝐴 ran 𝑥    ⇒   (𝜑 → ∩ 𝐴 = (𝑋 × 𝑌))
 
30-Oct-2025iinxp 48955 Indexed intersection of Cartesian products is the Cartesian product of indexed intersections. See also inxp 5775 and intxp 48956. (Contributed by Zhi Wang, 30-Oct-2025.)
(𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶))
 
30-Oct-2025iineq0 48944 An indexed intersection is empty if one of the intersected classes is empty. (Contributed by Zhi Wang, 30-Oct-2025.)
(∃𝑥 ∈ 𝐴 𝐵 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = ∅)
 
30-Oct-2025upgrimpthslem1 48031 Lemma 1 for upgrimpths 48033. (Contributed by AV, 30-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹(Paths‘𝐺)𝑃)    ⇒   (𝜑 → Fun ◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
 
30-Oct-20252f1fvneq 7200 If two one-to-one functions are applied on different arguments, also the values are different. (Contributed by Alexander van der Vekens, 25-Jan-2018.) (Proof shortened by AV, 30-Oct-2025.)
(((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → 𝑋 ≠ 𝑌))
 
30-Oct-2025dff14i 7199 A one-to-one function maps different arguments onto different values. Implication of the alternate definition dff14a 7210. (Contributed by AV, 30-Oct-2025.)
((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌))
 
29-Oct-2025upgrimtrls 48030 Graph isomorphisms between simple pseudographs map trails onto trails. (Contributed by AV, 29-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹(Trails‘𝐺)𝑃)    ⇒   (𝜑 → 𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃))
 
29-Oct-2025upgrimtrlslem2 48029 Lemma 2 for upgrimtrls 48030. (Contributed by AV, 29-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹(Trails‘𝐺)𝑃)    ⇒   ((𝜑 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹)) → ((◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))) = (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑦)))) → 𝑥 = 𝑦))
 
29-Oct-2025upgrimtrlslem1 48028 Lemma 1 for upgrimtrls 48030. (Contributed by AV, 29-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹(Trails‘𝐺)𝑃)    ⇒   ((𝜑 ∧ 𝑋 ∈ dom 𝐹) → (𝑁 “ (𝐼‘(𝐹‘𝑋))) ∈ (Edg‘𝐻))
 
28-Oct-2025upgrimwlk 48026 Graph isomorphisms between simple pseudographs map walks onto walks. (Contributed by AV, 28-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹(Walks‘𝐺)𝑃)    ⇒   (𝜑 → 𝐸(Walks‘𝐻)(𝑁 ∘ 𝑃))
 
28-Oct-2025upgrimwlklem5 48025 Lemma 5 for upgrimwlk 48026. (Contributed by AV, 28-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹(Walks‘𝐺)𝑃)    ⇒   ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐸))) → (𝑁 “ (𝐼‘(𝐹‘𝑖))) = {((𝑁 ∘ 𝑃)‘𝑖), ((𝑁 ∘ 𝑃)‘(𝑖 + 1))})
 
28-Oct-2025upgrimwlklem4 48024 Lemma 4 for upgrimwlk 48026. (Contributed by AV, 28-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹 ∈ Word dom 𝐼)    &   (𝜑 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))    ⇒   (𝜑 → (𝑁 ∘ 𝑃):(0...(♯‘𝐸))⟶(Vtx‘𝐻))
 
27-Oct-2025oppczeroo 49362 Zero objects are zero in the opposite category. Remark 7.8 of [Adamek] p. 103. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝐼 ∈ (ZeroO‘𝐶) ↔ 𝐼 ∈ (ZeroO‘(oppCat‘𝐶)))
 
27-Oct-2025oppcciceq 49177 The opposite category has the same isomorphic objects as the original category. (Contributed by Zhi Wang, 27-Oct-2025.)
𝑂 = (oppCat‘𝐶)    ⇒   ( ≃𝑐 ‘𝐶) = ( ≃𝑐 ‘𝑂)
 
27-Oct-2025oppccicb 49176 Isomorphic objects are isomorphic in the opposite category. (Contributed by Zhi Wang, 27-Oct-2025.)
𝑂 = (oppCat‘𝐶)    ⇒   (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ 𝑅( ≃𝑐 ‘𝑂)𝑆)
 
27-Oct-2025cicpropd 49175 Two structures with the same base, hom-sets and composition operation have the same isomorphic objects. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   (𝜑 → ( ≃𝑐 ‘𝐶) = ( ≃𝑐 ‘𝐷))
 
27-Oct-2025cicpropdlem 49174 Lemma for cicpropd 49175. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 ∈ ( ≃𝑐 ‘𝐷))
 
27-Oct-2025cic1st2ndbr 49173 Rewrite the predicate of isomorphic objects with separated parts. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝑃 ∈ ( ≃𝑐 ‘𝐶) → (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃))
 
27-Oct-2025cic1st2nd 49172 Reconstruction of a pair of isomorphic objects in terms of its ordered pair components. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
 
27-Oct-2025relcic 49170 The set of isomorphic objects is a relation. Simplifies cicer 17715 (see cicerALT 49171). (Contributed by Zhi Wang, 27-Oct-2025.)
(𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))
 
27-Oct-2025isopropd 49166 Two structures with the same base, hom-sets and composition operation have the same isomorphisms. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   (𝜑 → (Iso‘𝐶) = (Iso‘𝐷))
 
27-Oct-2025isopropdlem 49165 Lemma for isopropd 49166. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝑃 ∈ (Iso‘𝐷))
 
27-Oct-2025invpropd 49164 Two structures with the same base, hom-sets and composition operation have the same inverses. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   (𝜑 → (Inv‘𝐶) = (Inv‘𝐷))
 
27-Oct-2025invpropdlem 49163 Lemma for invpropd 49164. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → 𝑃 ∈ (Inv‘𝐷))
 
27-Oct-2025sectpropd 49162 Two structures with the same base, hom-sets and composition operation have the same sections. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   (𝜑 → (Sect‘𝐶) = (Sect‘𝐷))
 
27-Oct-2025sectpropdlem 49161 Lemma for sectpropd 49162. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   ((𝜑 ∧ 𝑃 ∈ (Sect‘𝐶)) → 𝑃 ∈ (Sect‘𝐷))
 
27-Oct-2025isofval2 49157 Function value of the function returning the isomorphisms of a category. (Contributed by Zhi Wang, 27-Oct-2025.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   𝐼 = (Iso‘𝐶)    ⇒   (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦)))
 
27-Oct-2025invfn 49155 The function value of the function returning the inverses of a category is a function over the Cartesian square of the base set of the category. Simplifies isofn 17684 (see isofnALT 49156). (Contributed by Zhi Wang, 27-Oct-2025.)
(𝐶 ∈ Cat → (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
 
27-Oct-2025sectfn 49154 The function value of the function returning the sections of a category is a function over the Cartesian square of the base set of the category. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝐶 ∈ Cat → (Sect‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
 
27-Oct-2025eloprab1st2nd 48992 Reconstruction of a nested ordered pair in terms of its ordered pair components. (Contributed by Zhi Wang, 27-Oct-2025.)
(𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝐴 = ⟨⟨(1st ‘(1st ‘𝐴)), (2nd ‘(1st ‘𝐴))⟩, (2nd ‘𝐴)⟩)
 
27-Oct-2025invffval 17667 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) Removed redundant hypotheses. (Revised by Zhi Wang, 27-Oct-2025.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   𝑆 = (Sect‘𝐶)    ⇒   (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))
 
27-Oct-2025sectffval 17659 Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.) Removed redundant hypotheses. (Revised by Zhi Wang, 27-Oct-2025.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &    1 = (Id‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    ⇒   (𝜑 → 𝑆 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))}))
 
26-Oct-2025termccisoeu 49642 The isomorphism between terminal categories is unique. (Contributed by Zhi Wang, 26-Oct-2025.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑋 ∈ TermCat)    &   (𝜑 → 𝑌 ∈ TermCat)    ⇒   (𝜑 → ∃!𝑓 𝑓 ∈ (𝑋(Iso‘𝐶)𝑌))
 
26-Oct-2025termcciso 49641 A category is isomorphic to a terminal category iff it itself is terminal. (Contributed by Zhi Wang, 26-Oct-2025.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑋 ∈ TermCat)    ⇒   (𝜑 → (𝑌 ∈ TermCat ↔ 𝑋( ≃𝑐 ‘𝐶)𝑌))
 
26-Oct-2025zeroopropd 49370 Two structures with the same base, hom-sets and composition operation have the same zero objects. (Contributed by Zhi Wang, 26-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷))
 
26-Oct-2025termopropd 49369 Two structures with the same base, hom-sets and composition operation have the same terminal objects. (Contributed by Zhi Wang, 26-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   (𝜑 → (TermO‘𝐶) = (TermO‘𝐷))
 
26-Oct-2025zeroopropdlem 49367 Lemma for zeroopropd 49370. (Contributed by Zhi Wang, 26-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   (𝜑 → ¬ 𝐶 ∈ V)    ⇒   (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷))
 
26-Oct-2025termopropdlem 49366 Lemma for termopropd 49369. (Contributed by Zhi Wang, 26-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   (𝜑 → ¬ 𝐶 ∈ V)    ⇒   (𝜑 → (TermO‘𝐶) = (TermO‘𝐷))
 
26-Oct-2025initopropdlem 49365 Lemma for initopropd 49368. (Contributed by Zhi Wang, 26-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   (𝜑 → ¬ 𝐶 ∈ V)    ⇒   (𝜑 → (InitO‘𝐶) = (InitO‘𝐷))
 
26-Oct-2025initopropdlemlem 49364 Lemma for initopropdlem 49365, termopropdlem 49366, and zeroopropdlem 49367. (Contributed by Zhi Wang, 26-Oct-2025.)
𝐹 Fn 𝑋    &   (𝜑 → ¬ 𝐴 ∈ 𝑌)    &   𝑋 ⊆ 𝑌    &   ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) = ∅)    ⇒   (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))
 
26-Oct-2025termoeu2 49363 Terminal objects are essentially unique; if 𝐴 is a terminal object, then so is every object that is isomorphic to 𝐴. (Contributed by Zhi Wang, 26-Oct-2025.)
(𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐴 ∈ (TermO‘𝐶))    &   (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)    ⇒   (𝜑 → 𝐵 ∈ (TermO‘𝐶))
 
26-Oct-2025oppctermo 49361 Terminal objects are initial in the opposite category. Comments before Definition 7.4 in [Adamek] p. 102. (Contributed by Zhi Wang, 26-Oct-2025.)
(𝐼 ∈ (TermO‘𝐶) ↔ 𝐼 ∈ (InitO‘(oppCat‘𝐶)))
 
26-Oct-2025oppccic 49169 Isomorphic objects are isomorphic in the opposite category. (Contributed by Zhi Wang, 26-Oct-2025.)
𝑂 = (oppCat‘𝐶)    &   (𝜑 → 𝑅( ≃𝑐 ‘𝐶)𝑆)    ⇒   (𝜑 → 𝑅( ≃𝑐 ‘𝑂)𝑆)
 
26-Oct-2025cicrcl2 49168 Isomorphism implies the structure being a category. (Contributed by Zhi Wang, 26-Oct-2025.)
(𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝐶 ∈ Cat)
 
26-Oct-2025cicfn 49167 ≃𝑐 is a function on Cat. (Contributed by Zhi Wang, 26-Oct-2025.)
≃𝑐 Fn Cat
 
26-Oct-2025tcfr 45080 A set is well-founded if and only if its transitive closure is well-founded by ∈. This characterization of well-founded sets is that in Definition I.9.20 of [Kunen2] p. 53. (Contributed by Eric Schmidt, 26-Oct-2025.)
𝐴 ∈ V    ⇒   (𝐴 ∈ ∪ (𝑅1 “ On) ↔ E Fr (TC‘𝐴))
 
26-Oct-2025trfr 45079 A transitive class well-founded by ∈ is a subclass of the class of well-founded sets. Part of Lemma I.9.21 of [Kunen2] p. 53. (Contributed by Eric Schmidt, 26-Oct-2025.)
((Tr 𝐴 ∧ E Fr 𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))
 
26-Oct-20252sqr3nconstr 33815 Doubling the cube is an impossible construction, i.e. the cube root of 2 is not constructible with straightedge and compass. Given a cube of edge of length one, a cube of double volume would have an edge of length (2↑𝑐(1 / 3)), however that number is not constructible. This is the first part of Metamath 100 proof #8. Theorem 7.13 of [Stewart] p. 99. (Contributed by Thierry Arnoux and Saveliy Skresanov, 26-Oct-2025.)
(2↑𝑐(1 / 3)) ∉ Constr
 
26-Oct-2025constrcon 33808 Contradiction of constructibility: If a complex number 𝐴 has minimal polynomial 𝐹 over ℚ of a degree that is not a power of 2, then 𝐴 is not constructible. (Contributed by Thierry Arnoux, 26-Oct-2025.)
𝐷 = (deg1‘(ℂfld ↾s ℚ))    &   𝑀 = (ℂfld minPoly ℚ)    &   (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐹 = (𝑀‘𝐴))    &   (𝜑 → (𝐷‘𝐹) ∈ ℕ0)    &   ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐷‘𝐹) ≠ (2↑𝑛))    ⇒   (𝜑 → ¬ 𝐴 ∈ Constr)
 
26-Oct-2025constrext2chn 33793 If a constructible number generates some subfield 𝐿 of ℂ, then the degree of the extension of 𝐿 over ℚ is a power of two. Theorem 7.12 of [Stewart] p. 98. (Contributed by Thierry Arnoux, 26-Oct-2025.)
𝑄 = (ℂfld ↾s ℚ)    &   𝐿 = (ℂfld ↾s 𝑆)    &   𝑆 = (ℂfld fldGen (ℚ ∪ {𝐴}))    &   (𝜑 → 𝐴 ∈ Constr)    ⇒   (𝜑 → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
 
26-Oct-2025constrext2chnlem 33784 Lemma for constrext2chn 33793. (Contributed by Thierry Arnoux, 26-Oct-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   𝐸 = (ℂfld ↾s 𝑒)    &   𝐹 = (ℂfld ↾s 𝑓)    &    < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}    &   (𝜑 → 𝑁 ∈ ω)    &   𝑄 = (ℂfld ↾s ℚ)    &   𝐿 = (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))    &   (𝜑 → 𝐴 ∈ Constr)    ⇒   (𝜑 → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
 
26-Oct-2025minplyelirng 33749 If the minimial polynomial 𝐹 of an element 𝑋 of a field 𝑅 has nonnegative degree, then 𝑋 is integral. (Contributed by Thierry Arnoux, 26-Oct-2025.)
𝐵 = (Base‘𝑅)    &   𝑀 = (𝑅 minPoly 𝑆)    &   𝐷 = (deg1‘(𝑅 ↾s 𝑆))    &   (𝜑 → 𝑅 ∈ Field)    &   (𝜑 → 𝑆 ∈ (SubDRing‘𝑅))    &   (𝜑 → 𝐴 ∈ 𝐵)    &   (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)    ⇒   (𝜑 → 𝐴 ∈ (𝑅 IntgRing 𝑆))
 
26-Oct-2025minplynzm1p 33748 If a minimal polynomial is nonzero, then it is monic. (Contributed by Thierry Arnoux, 26-Oct-2025.)
𝐵 = (Base‘𝐸)    &   𝑍 = (0g‘(Poly1‘𝐸))    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   𝑀 = (𝐸 minPoly 𝐹)    &   (𝜑 → 𝐴 ∈ 𝐵)    &   (𝜑 → (𝑀‘𝐴) ≠ 𝑍)    &   𝑈 = (Monic1p‘(𝐸 ↾s 𝐹))    ⇒   (𝜑 → (𝑀‘𝐴) ∈ 𝑈)
 
26-Oct-2025fldextsdrg 33688 Deduce sub-division-ring from field extension. (Contributed by Thierry Arnoux, 26-Oct-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑 → 𝐸/FldExt𝐹)    ⇒   (𝜑 → 𝐵 ∈ (SubDRing‘𝐸))
 
26-Oct-2025sdrgfldext 33684 A field 𝐸 and any sub-division-ring 𝐹 of 𝐸 form a field extension. (Contributed by Thierry Arnoux, 26-Oct-2025.)
𝐵 = (Base‘𝐸)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    ⇒   (𝜑 → 𝐸/FldExt(𝐸 ↾s 𝐹))
 
26-Oct-2025subsdrg 33271 A subring of a sub-division-ring is a sub-division-ring. See also subsubrg 20515. (Contributed by Thierry Arnoux, 26-Oct-2025.)
𝑆 = (𝑅 ↾s 𝐴)    &   (𝜑 → 𝐴 ∈ (SubDRing‘𝑅))    ⇒   (𝜑 → (𝐵 ∈ (SubDRing‘𝑆) ↔ (𝐵 ∈ (SubDRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)))
 
26-Oct-2025hashne0 32797 Deduce that the size of a set is not zero. (Contributed by Thierry Arnoux, 26-Oct-2025.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐴 ≠ ∅)    ⇒   (𝜑 → 0 < (♯‘𝐴))
 
26-Oct-2025xnn0nnd 32760 Conditions for an extended nonnegative integer to be a positive integer. (Contributed by Thierry Arnoux, 26-Oct-2025.)
(𝜑 → 𝑁 ∈ ℕ0*)    &   (𝜑 → 𝑁 ∈ ℝ)    &   (𝜑 → 0 < 𝑁)    ⇒   (𝜑 → 𝑁 ∈ ℕ)
 
26-Oct-2025xnn0nn0d 32759 Conditions for an extended nonnegative integer to be a nonnegative integer. (Contributed by Thierry Arnoux, 26-Oct-2025.)
(𝜑 → 𝑁 ∈ ℕ0*)    &   (𝜑 → 𝑁 ∈ ℝ)    ⇒   (𝜑 → 𝑁 ∈ ℕ0)
 
26-Oct-2025rexmul2 32741 If the result 𝐴 of an extended real multiplication is real, then its first factor 𝐵 is also real. See also rexmul 13172. (Contributed by Thierry Arnoux, 26-Oct-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ*)    &   (𝜑 → 𝐶 ∈ ℝ*)    &   (𝜑 → 0 < 𝐶)    &   (𝜑 → 𝐴 = (𝐵 ·e 𝐶))    ⇒   (𝜑 → 𝐵 ∈ ℝ)
 
26-Oct-2025ee4anv 2353 Distribute two pairs of existential quantifiers over a conjunction. For a version requiring fewer axioms but with additional disjoint variable conditions, see 4exdistrv 1957. (Contributed by NM, 31-Jul-1995.) Remove disjoint variable conditions on 𝑦, 𝑧 and 𝑥, 𝑤. (Revised by Eric Schmidt, 26-Oct-2025.)
(∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))
 
25-Oct-2025upgrimwlklem3 48023 Lemma 3 for upgrimwlk 48026. (Contributed by AV, 25-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹 ∈ Word dom 𝐼)    ⇒   ((𝜑 ∧ 𝑋 ∈ (0..^(♯‘𝐸))) → (𝐽‘(𝐸‘𝑋)) = (𝑁 “ (𝐼‘(𝐹‘𝑋))))
 
25-Oct-2025upgrimwlklem2 48022 Lemma 2 for upgrimwlk 48026. (Contributed by AV, 25-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹 ∈ Word dom 𝐼)    ⇒   (𝜑 → 𝐸 ∈ Word dom 𝐽)
 
25-Oct-2025upgrimwlklem1 48021 Lemma 1 for upgrimwlk 48026 and upgrimwlklen 48027. (Contributed by AV, 25-Oct-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   (𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   (𝜑 → 𝐹 ∈ Word dom 𝐼)    ⇒   (𝜑 → (♯‘𝐸) = (♯‘𝐹))
 
25-Oct-2025uhgrimprop 48016 An isomorphism between hypergraphs is a bijection between their vertices that preserves adjacency for simple edges, i.e. there is a simple edge in one graph connecting one or two vertices iff there is a simple edge in the other graph connecting the vertices which are the images of the vertices. (Contributed by AV, 27-Apr-2025.) (Revised by AV, 25-Oct-2025.)
𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    ⇒   ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)))
 
25-Oct-2025uhgrimedg 48015 An isomorphism between graphs preserves edges, i.e. there is an edge in one graph connecting vertices iff there is an edge in the other graph connecting the corresponding vertices. (Contributed by AV, 25-Oct-2025.)
𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)    ⇒   (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐾 ∈ 𝐸 ↔ (𝐹 “ 𝐾) ∈ 𝐷))
 
25-Oct-2025uhgrimedgi 48014 An isomorphism between graphs preserves edges, i.e. if there is an edge in one graph connecting vertices then there is an edge in the other graph connecting the corresponding vertices. (Contributed by AV, 25-Oct-2025.)
𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)    ⇒   (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ (𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸)) → (𝐹 “ 𝐾) ∈ 𝐷)
 
23-Oct-2025termcterm2 49639 A terminal object of the category of small categories is a terminal category. (Contributed by Zhi Wang, 18-Oct-2025.) (Proof shortened by Zhi Wang, 23-Oct-2025.)
𝐸 = (CatCat‘𝑈)    &   (𝜑 → (𝑈 ∩ TermCat) ≠ ∅)    &   (𝜑 → 𝐶 ∈ (TermO‘𝐸))    ⇒   (𝜑 → 𝐶 ∈ TermCat)
 
23-Oct-2025dftermo4 49627 An alternate definition of df-termo 17894 using universal property. See also the "Equivalent formulations" section of https://en.wikipedia.org/wiki/Initial_and_terminal_objects 17894. (Contributed by Zhi Wang, 23-Oct-2025.)
TermO = (𝑐 ∈ Cat ↦ ⦋(oppCat‘𝑐) / 𝑜⦌⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑜))‘∅) / 𝑓⦌dom (𝑓(𝑜 UP 𝑑)∅))
 
23-Oct-2025dfinito4 49626 An alternate definition of df-inito 17893 using universal property. See also the "Equivalent formulations" section of https://en.wikipedia.org/wiki/Initial_and_terminal_objects 17893. (Contributed by Zhi Wang, 23-Oct-2025.)
InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))
 
23-Oct-2025isinito3 49625 The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 23-Oct-2025.)
1 = (SetCat‘1o)    &   𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)    ⇒   (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ dom (𝐹(𝐶 UP 1 )∅))
 
23-Oct-2025isinito2 49624 The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 23-Oct-2025.)
1 = (SetCat‘1o)    &   𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)    ⇒   (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼(𝐹(𝐶 UP 1 )∅)∅)
 
23-Oct-2025isinito2lem 49623 The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 23-Oct-2025.)
1 = (SetCat‘1o)    &   𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐼 ∈ (Base‘𝐶))    ⇒   (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼(𝐹(𝐶 UP 1 )∅)∅))
 
23-Oct-2025initopropd 49368 Two structures with the same base, hom-sets and composition operation have the same initial objects. (Contributed by Zhi Wang, 23-Oct-2025.)
(𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   (𝜑 → (InitO‘𝐶) = (InitO‘𝐷))
 
23-Oct-2025oppcinito 49360 Initial objects are terminal in the opposite category. (Contributed by Zhi Wang, 23-Oct-2025.)
(𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ (TermO‘(oppCat‘𝐶)))
 
23-Oct-2025zeroo2 49359 A zero object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.)
𝐵 = (Base‘𝐶)    ⇒   (𝑂 ∈ (ZeroO‘𝐶) → 𝑂 ∈ 𝐵)
 
23-Oct-2025termoo2 49358 A terminal object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.)
𝐵 = (Base‘𝐶)    ⇒   (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵)
 
23-Oct-2025initoo2 49357 An initial object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.)
𝐵 = (Base‘𝐶)    ⇒   (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ 𝐵)
 
23-Oct-2025up1st2nd2 49313 Rewrite the universal property predicate with separated parts. (Contributed by Zhi Wang, 23-Oct-2025.)
(𝜑 → 𝑋 ∈ (𝐹(𝐷 UP 𝐸)𝑊))    ⇒   (𝜑 → (1st ‘𝑋)(𝐹(𝐷 UP 𝐸)𝑊)(2nd ‘𝑋))
 
23-Oct-2025up1st2ndb 49312 Combine/separate parts in the universal property predicate. (Contributed by Zhi Wang, 23-Oct-2025.)
(𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    ⇒   (𝜑 → (𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀 ↔ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀))
 
23-Oct-2025up1st2ndr 49311 Combine separated parts in the universal property predicate. (Contributed by Zhi Wang, 23-Oct-2025.)
(𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)    ⇒   (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)
 
23-Oct-2025up1st2nd 49310 Rewrite the universal property predicate with separated parts. (Contributed by Zhi Wang, 23-Oct-2025.)
(𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)    ⇒   (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)
 
23-Oct-2025oppccatb 49141 An opposite category is a category. (Contributed by Zhi Wang, 23-Oct-2025.)
𝑂 = (oppCat‘𝐶)    &   (𝜑 → 𝐶 ∈ 𝑉)    ⇒   (𝜑 → (𝐶 ∈ Cat ↔ 𝑂 ∈ Cat))
 
23-Oct-2025homf0 49134 The base is empty iff the functionalized Hom-set operation is empty. (Contributed by Zhi Wang, 23-Oct-2025.)
((Base‘𝐶) = ∅ ↔ (Homf ‘𝐶) = ∅)
 
22-Oct-2025mndtchom 49709 The only hom-set of the category built from a monoid is the base set of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (Proof shortened by Zhi Wang, 22-Oct-2025.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   (𝜑 → (𝑋𝐻𝑌) = (Base‘𝑀))
 
22-Oct-2025funcsetc1o 49622 Value of the functor to the trivial category. The converse is also true because 𝐹 would be the empty set if 𝐶 were not a category; and the empty set cannot equal an ordered pair of two sets. (Contributed by Zhi Wang, 22-Oct-2025.)
1 = (SetCat‘1o)    &   𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)    &   (𝜑 → 𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝜑 → 𝐹 = ⟨(𝐵 × 1o), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × 1o))⟩)
 
22-Oct-2025funcsetc1ocl 49621 The functor to the trivial category. The converse is also true due to reverse closure. (Contributed by Zhi Wang, 22-Oct-2025.)
1 = (SetCat‘1o)    &   𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)    &   (𝜑 → 𝐶 ∈ Cat)    ⇒   (𝜑 → 𝐹 ∈ (𝐶 Func 1 ))
 
22-Oct-2025setc1oid 49620 The identity morphism of the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)
1 = (SetCat‘1o)    &   𝐼 = (Id‘ 1 )    ⇒   (𝐼‘∅) = ∅
 
22-Oct-2025setc1ocofval 49619 Composition in the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)
1 = (SetCat‘1o)    ⇒   {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )
 
22-Oct-2025setc1ohomfval 49618 Set of morphisms of the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)
1 = (SetCat‘1o)    ⇒   {⟨∅, ∅, 1o⟩} = (Hom ‘ 1 )
 
22-Oct-2025setc1obas 49617 The base of the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)
1 = (SetCat‘1o)    ⇒   1o = (Base‘ 1 )
 
22-Oct-2025ovsn2 48985 The operation value of a singleton of an ordered triple is the last member. (Contributed by Zhi Wang, 22-Oct-2025.)
𝐶 ∈ V    ⇒   (𝐴{⟨𝐴, 𝐵, 𝐶⟩}𝐵) = 𝐶
 
22-Oct-2025ovsn 48984 The operation value of a singleton of a nested ordered pair is the last member. (Contributed by Zhi Wang, 22-Oct-2025.)
𝐶 ∈ V    ⇒   (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = 𝐶
 
22-Oct-2025ovsng2 48983 The operation value of a singleton of an ordered triple is the last member. (Contributed by Zhi Wang, 22-Oct-2025.)
(𝐶 ∈ 𝑉 → (𝐴{⟨𝐴, 𝐵, 𝐶⟩}𝐵) = 𝐶)
 
22-Oct-2025ovsng 48982 The operation value of a singleton of a nested ordered pair is the last member. (Contributed by Zhi Wang, 22-Oct-2025.)
(𝐶 ∈ 𝑉 → (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = 𝐶)
 
22-Oct-2025dfbi1ALTb 45058 Further shorten dfbi1ALTa 45056 using simprimi 45057. (Contributed by Eric Schmidt, 22-Oct-2025.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑 ↔ 𝜓) ↔ ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
 
22-Oct-2025simprimi 45057 Inference associated with simprim 166. Proved exactly as step 11 is obtained from step 4 in dfbi1ALTa 45056. (Contributed by Eric Schmidt, 22-Oct-2025.) (New usage is discouraged.) (Proof modification is discouraged.)
¬ (𝜑 → ¬ 𝜓)    ⇒   𝜓
 
22-Oct-2025dfbi1ALTa 45056 Version of dfbi1ALT 214 using ⊤ for step 2 and shortened using a1i 11, a2i 14, and con4i 114. (Contributed by Eric Schmidt, 22-Oct-2025.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑 ↔ 𝜓) ↔ ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
 
22-Oct-20259ne0 42382 The number 9 is nonzero. (Contributed by SN, 22-Oct-2025.)
9 ≠ 0
 
22-Oct-20258ne0 42381 The number 8 is nonzero. (Contributed by SN, 22-Oct-2025.)
8 ≠ 0
 
22-Oct-20257ne0 42380 The number 7 is nonzero. (Contributed by SN, 22-Oct-2025.)
7 ≠ 0
 
22-Oct-20256ne0 42379 The number 6 is nonzero. (Contributed by SN, 22-Oct-2025.)
6 ≠ 0
 
22-Oct-20255ne0 42378 The number 5 is nonzero. (Contributed by SN, 22-Oct-2025.)
5 ≠ 0
 
22-Oct-2025halfpm6th 12350 One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.) (Proof shortened by SN, 22-Oct-2025.)
(((1 / 2) − (1 / 6)) = (1 / 3) ∧ ((1 / 2) + (1 / 6)) = (2 / 3))
 
22-Oct-20251mhlfehlf 12347 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.) (Proof shortened by SN, 22-Oct-2025.)
(1 − (1 / 2)) = (1 / 2)
 
21-Oct-2025diagcic 49665 Any category 𝐶 is isomorphic to the category of functors from a terminal category to 𝐶. See also the "Properties" section of https://ncatlab.org/nlab/show/terminal+category. Therefore the number of categories isomorphic to a non-empty category is at least the number of singletons, so large (snnex 7697) that these isomorphic categories form a proper class. (Contributed by Zhi Wang, 21-Oct-2025.)
(𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ TermCat)    &   𝑄 = (𝐷 FuncCat 𝐶)    &   𝐸 = (CatCat‘𝑈)    &   (𝜑 → 𝑈 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑈)    &   (𝜑 → 𝑄 ∈ 𝑈)    ⇒   (𝜑 → 𝐶( ≃𝑐 ‘𝐸)𝑄)
 
21-Oct-2025diagciso 49664 The diagonal functor is an isomorphism from a category 𝐶 to the category of functors from a terminal category to 𝐶.

It is provable that the inverse of the diagonal functor is the mapped object by the transposed curry of (𝐷 evalF 𝐶), i.e., ∪ ran (1st ‘(⟨𝐷, 𝑄⟩ curryF ((𝐷 evalF 𝐶) ∘func (𝐷 swapF 𝑄)))).

(Contributed by Zhi Wang, 21-Oct-2025.)

(𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ TermCat)    &   𝑄 = (𝐷 FuncCat 𝐶)    &   𝐸 = (CatCat‘𝑈)    &   (𝜑 → 𝑈 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑈)    &   (𝜑 → 𝑄 ∈ 𝑈)    &   𝐼 = (Iso‘𝐸)    &   𝐿 = (𝐶Δfunc𝐷)    ⇒   (𝜑 → 𝐿 ∈ (𝐶𝐼𝑄))
 
21-Oct-2025diagffth 49663 The diagonal functor is a fully faithful functor from a category 𝐶 to the category of functors from a terminal category to 𝐶. (Contributed by Zhi Wang, 21-Oct-2025.)
(𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ TermCat)    &   𝑄 = (𝐷 FuncCat 𝐶)    &   𝐿 = (𝐶Δfunc𝐷)    ⇒   (𝜑 → 𝐿 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
 
21-Oct-2025diag2f1o 49662 If 𝐷 is terminal, the morphism part of a diagonal functor is bijective functions from hom-sets into sets of natural transformations. (Contributed by Zhi Wang, 21-Oct-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   𝐴 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   𝑁 = (𝐷 Nat 𝐶)    &   (𝜑 → 𝐷 ∈ TermCat)    &   (𝜑 → 𝐶 ∈ Cat)    ⇒   (𝜑 → (𝑋(2nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1-onto→(((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑌)))
 
21-Oct-2025diag2f1olem 49661 Lemma for diag2f1o 49662. (Contributed by Zhi Wang, 21-Oct-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   𝐴 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   𝑁 = (𝐷 Nat 𝐶)    &   (𝜑 → 𝐷 ∈ TermCat)    &   (𝜑 → 𝑀 ∈ (((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑌)))    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   𝐹 = (𝑀‘𝑍)    ⇒   (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑀 = ((𝑋(2nd ‘𝐿)𝑌)‘𝐹)))
 
21-Oct-2025termcnatval 49660 Value of natural transformations for a terminal category. (Contributed by Zhi Wang, 21-Oct-2025.)
(𝜑 → 𝐶 ∈ TermCat)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   𝑅 = (𝐴‘𝑋)    ⇒   (𝜑 → 𝐴 = {⟨𝑋, 𝑅⟩})
 
21-Oct-2025diag1f1o 49659 The object part of the diagonal functor is a bijection if 𝐷 is terminal. So any functor from a terminal category is one-to-one correspondent to an object of the target base. (Contributed by Zhi Wang, 21-Oct-2025.)
𝐴 = (Base‘𝐶)    &   (𝜑 → 𝐷 ∈ TermCat)    &   (𝜑 → 𝐶 ∈ Cat)    &   𝐿 = (𝐶Δfunc𝐷)    ⇒   (𝜑 → (1st ‘𝐿):𝐴–1-1-onto→(𝐷 Func 𝐶))
 
21-Oct-2025diag1f1olem 49658 To any functor from a terminal category can an object in the target base be assigned. (Contributed by Zhi Wang, 21-Oct-2025.)
𝐴 = (Base‘𝐶)    &   (𝜑 → 𝐷 ∈ TermCat)    &   (𝜑 → 𝐾 ∈ (𝐷 Func 𝐶))    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑋 = ((1st ‘𝐾)‘𝑌)    &   𝐿 = (𝐶Δfunc𝐷)    ⇒   (𝜑 → (𝑋 ∈ 𝐴 ∧ 𝐾 = ((1st ‘𝐿)‘𝑋)))
 
21-Oct-2025termchom2 49614 The hom-set of a terminal category is a singleton of the identity morphism. (Contributed by Zhi Wang, 21-Oct-2025.)
(𝜑 → 𝐶 ∈ TermCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐻 = (Hom ‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑍)})
 
21-Oct-2025diag2f1 49434 If 𝐵 is non-empty, the morphism part of a diagonal functor is injective functions from hom-sets into sets of natural transformations. (Contributed by Zhi Wang, 21-Oct-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   𝐴 = (Base‘𝐶)    &   𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   (𝜑 → 𝐵 ≠ ∅)    &   𝑁 = (𝐷 Nat 𝐶)    ⇒   (𝜑 → (𝑋(2nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑌)))
 
21-Oct-2025diag2f1lem 49433 Lemma for diag2f1 49434. The converse is trivial (fveq2 6828). (Contributed by Zhi Wang, 21-Oct-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   𝐴 = (Base‘𝐶)    &   𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   (𝜑 → 𝐵 ≠ ∅)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    ⇒   (𝜑 → (((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘𝐿)𝑌)‘𝐺) → 𝐹 = 𝐺))
 
21-Oct-2025fnsnb 7105 A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.) (Proof shortened by Zhi Wang, 21-Oct-2025.)
𝐴 ∈ V    ⇒   (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})
 
21-Oct-2025fnsnbg 7104 A function's domain is a singleton iff the function is a singleton. (Contributed by Steven Nguyen, 18-Aug-2023.) Relax condition for being in the universal class. (Revised by Zhi Wang, 21-Oct-2025.)
(𝐴 ∈ 𝑉 → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))
 
20-Oct-2025termcnex 49701 The class of all terminal categories is a proper class. Therefore both the class of all thin categories and the class of all categories are proper classes. Note that snnex 7697 is equivalent to sngl V ∉ V. (Contributed by Zhi Wang, 20-Oct-2025.)
TermCat ∉ V
 
20-Oct-2025basrestermcfo 49700 The base function restricted to the class of terminal categories maps the class of terminal categories onto the class of singletons. (Contributed by Zhi Wang, 20-Oct-2025.)
(Base ↾ TermCat):TermCat–onto→{𝑏 ∣ ∃𝑥 𝑏 = {𝑥}}
 
20-Oct-2025discsnterm 49699 A discrete category (a category whose only morphisms are the identity morphisms) with a singlegon base is terminal. Corollary of example 3.3(4)(c) of [Adamek] p. 24 and example 3.26(1) of [Adamek] p. 33. (Contributed by Zhi Wang, 20-Oct-2025.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    &   𝐶 = (ProsetToCat‘𝐾)    ⇒   (∃𝑥 𝐵 = {𝑥} → 𝐶 ∈ TermCat)
 
20-Oct-2025discthin 49698 A discrete category (a category whose only morphisms are the identity morphisms) is thin. (Contributed by Zhi Wang, 20-Oct-2025.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    &   𝐶 = (ProsetToCat‘𝐾)    ⇒   (𝐵 ∈ 𝑉 → 𝐶 ∈ ThinCat)
 
20-Oct-2025discbas 49697 A discrete category (a category whose only morphisms are the identity morphisms) can be constructed for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    &   𝐶 = (ProsetToCat‘𝐾)    ⇒   (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐶))
 
20-Oct-2025basrestermcfolem 49696 An element of the class of singlegons is a singlegon. The converse (discsntermlem 49695) also holds. This is trivial if 𝐵 is 𝑏 (abid 2715). (Contributed by Zhi Wang, 20-Oct-2025.)
(𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} → ∃𝑥 𝐵 = {𝑥})
 
20-Oct-2025discsntermlem 49695 A singlegon is an element of the class of singlegons. The converse (basrestermcfolem 49696) also holds. This is trivial if 𝐵 is 𝑏 (abid 2715). (Contributed by Zhi Wang, 20-Oct-2025.)
(∃𝑥 𝐵 = {𝑥} → 𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}})
 
20-Oct-2025termcfuncval 49657 The value of a functor from a terminal category. (Contributed by Zhi Wang, 20-Oct-2025.)
𝐴 = (Base‘𝐶)    &   (𝜑 → 𝐷 ∈ TermCat)    &   (𝜑 → 𝐾 ∈ (𝐷 Func 𝐶))    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑋 = ((1st ‘𝐾)‘𝑌)    &    1 = (Id‘𝐶)    &   𝐼 = (Id‘𝐷)    ⇒   (𝜑 → (𝑋 ∈ 𝐴 ∧ 𝐾 = ⟨{⟨𝑌, 𝑋⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩}⟩))
 
20-Oct-2025dftermc3 49656 Alternate definition of TermCat. See also df-termc 49598, dftermc2 49645. (Contributed by Zhi Wang, 20-Oct-2025.)
TermCat = {𝑐 ∣ (Arrow‘𝑐) ≈ 1o}
 
20-Oct-2025arweutermc 49655 If a structure has a unique disjointified arrow, then the structure is a terminal category. (Contributed by Zhi Wang, 20-Oct-2025.)
(∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ TermCat)
 
20-Oct-2025arweuthinc 49654 If a structure has a unique disjointified arrow, then the structure is a thin category. (Contributed by Zhi Wang, 20-Oct-2025.)
(∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ ThinCat)
 
20-Oct-2025termcarweu 49653 There exists a unique disjointified arrow in a terminal category. (Contributed by Zhi Wang, 20-Oct-2025.)
(𝐶 ∈ TermCat → ∃!𝑎 𝑎 ∈ (Arrow‘𝐶))
 
20-Oct-2025euendfunc2 49652 If there exists a unique endofunctor (a functor from a category to itself) for a category, then it is either initial (empty) or terminal. (Contributed by Zhi Wang, 20-Oct-2025.)
((𝐶 Func 𝐶) ≈ 1o → ((Base‘𝐶) = ∅ ∨ 𝐶 ∈ TermCat))
 
20-Oct-2025euendfunc 49651 If there exists a unique endofunctor (a functor from a category to itself) for a non-empty category, then the category is terminal. This partially explains why two categories are sufficient in termc2 49643. (Contributed by Zhi Wang, 20-Oct-2025.)
(𝜑 → ∃!𝑓 𝑓 ∈ (𝐶 Func 𝐶))    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝐵 ≠ ∅)    ⇒   (𝜑 → 𝐶 ∈ TermCat)
 
20-Oct-2025termc2 49643 If there exists a unique functor from both the category itself and the trivial category, then the category is terminal. Note that the converse also holds, so that it is a biconditional. See the proof of termc 49644 for hints. See also eufunc 49647 and euendfunc2 49652 for some insights on why two categories are sufficient. (Contributed by Zhi Wang, 18-Oct-2025.) (Proof shortened by Zhi Wang, 20-Oct-2025.)
(∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → 𝐶 ∈ TermCat)
 
20-Oct-2025termchom 49613 The hom-set of a terminal category is a singleton of the identity morphism. (Contributed by Zhi Wang, 20-Oct-2025.)
(𝜑 → 𝐶 ∈ TermCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐻 = (Hom ‘𝐶)    &    1 = (Id‘𝐶)    ⇒   (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})
 
20-Oct-2025termcbas2 49607 The base of a terminal category is given by its object. (Contributed by Zhi Wang, 20-Oct-2025.)
(𝜑 → 𝐶 ∈ TermCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝐵 = {𝑋})
 
20-Oct-2025thinchom 49552 A non-empty hom-set of a thin category is given by its element. (Contributed by Zhi Wang, 20-Oct-2025.)
(𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐶 ∈ ThinCat)    ⇒   (𝜑 → (𝑋𝐻𝑌) = {𝐹})
 
20-Oct-2025precoffunc 49497 The pre-composition functor, expressed explicitly, is a functor. (Contributed by Zhi Wang, 11-Oct-2025.) (Proof shortened by Zhi Wang, 20-Oct-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   𝐵 = (𝐷 Func 𝐸)    &   𝑁 = (𝐷 Nat 𝐸)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   (𝜑 → 𝐸 ∈ Cat)    &   (𝜑 → 𝐾 = (𝑔 ∈ 𝐵 ↦ (𝑔 ∘func ⟨𝐹, 𝐺⟩)))    &   (𝜑 → 𝐿 = (𝑔 ∈ 𝐵, ℎ ∈ 𝐵 ↦ (𝑎 ∈ (𝑔𝑁ℎ) ↦ (𝑎 ∘ 𝐹))))    &   𝑆 = (𝐶 FuncCat 𝐸)    ⇒   (𝜑 → 𝐾(𝑅 Func 𝑆)𝐿)
 
20-Oct-2025precofval3 49496 Value of the pre-composition functor as a transposed curry of the functor composition bifunctor. (Contributed by Zhi Wang, 20-Oct-2025.)
𝑅 = (𝐷 FuncCat 𝐸)    &   𝐵 = (𝐷 Func 𝐸)    &   𝑁 = (𝐷 Nat 𝐸)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   (𝜑 → 𝐸 ∈ Cat)    &   (𝜑 → 𝐾 = (𝑔 ∈ 𝐵 ↦ (𝑔 ∘func ⟨𝐹, 𝐺⟩)))    &   (𝜑 → 𝐿 = (𝑔 ∈ 𝐵, ℎ ∈ 𝐵 ↦ (𝑎 ∈ (𝑔𝑁ℎ) ↦ (𝑎 ∘ 𝐹))))    &   𝑄 = (𝐶 FuncCat 𝐷)    &   (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))    &   (𝜑 → 𝑀 = ((1st ‘ ⚬ )‘⟨𝐹, 𝐺⟩))    ⇒   (𝜑 → ⟨𝐾, 𝐿⟩ = 𝑀)
 
20-Oct-2025prsnex 49105 The class of preordered sets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025.)
Proset ∉ V
 
20-Oct-2025posnex 49104 The class of posets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025.)
Poset ∉ V
 
20-Oct-2025basresprsfo 49103 The base function restricted to the class of preordered sets maps the class of preordered sets onto the universal class. (Contributed by Zhi Wang, 20-Oct-2025.)
(Base ↾ Proset ): Proset –onto→V
 
20-Oct-2025basresposfo 49102 The base function restricted to the class of posets maps the class of posets onto the universal class. (Contributed by Zhi Wang, 20-Oct-2025.)
(Base ↾ Poset):Poset–onto→V
 
20-Oct-2025exbasprs 49101 There exists a preordered set for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
(𝐵 ∈ 𝑉 → ∃𝑘 ∈ Proset 𝐵 = (Base‘𝑘))
 
20-Oct-2025exbaspos 49100 There exists a poset for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
(𝐵 ∈ 𝑉 → ∃𝑘 ∈ Poset 𝐵 = (Base‘𝑘))
 
20-Oct-2025resipos 49099 A set equipped with an order where no distinct elements are comparable is a poset. (Contributed by Zhi Wang, 20-Oct-2025.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    ⇒   (𝐵 ∈ 𝑉 → 𝐾 ∈ Poset)
 
20-Oct-2025resiposbas 49098 Construct a poset (resipos 49099) for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    ⇒   (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐾))
 
20-Oct-2025slotresfo 49023 The condition of a structure component extractor restricted to a class being a surjection. This combined with fonex 48991 can be used to prove a class being proper. (Contributed by Zhi Wang, 20-Oct-2025.)
𝐸 Fn V    &   (𝑘 ∈ 𝐴 → (𝐸‘𝑘) ∈ 𝑉)    &   (𝑏 ∈ 𝑉 → 𝐾 ∈ 𝐴)    &   (𝑏 ∈ 𝑉 → 𝑏 = (𝐸‘𝐾))    ⇒   (𝐸 ↾ 𝐴):𝐴–onto→𝑉
 
20-Oct-2025fonex 48991 The domain of a surjection is a proper class if the range is a proper class as well. Can be used to prove that if a structure component extractor restricted to a class maps onto a proper class, then the class is a proper class as well. (Contributed by Zhi Wang, 20-Oct-2025.)
𝐵 ∉ V    &   𝐹:𝐴–onto→𝐵    ⇒   𝐴 ∉ V
 
19-Oct-2025idfudiag1 49650 If the identity functor of a category is the same as a constant functor to the category, then the category is terminal. (Contributed by Zhi Wang, 19-Oct-2025.)
𝐼 = (idfunc‘𝐶)    &   𝐿 = (𝐶Δfunc𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   𝐾 = ((1st ‘𝐿)‘𝑋)    &   (𝜑 → 𝐼 = 𝐾)    ⇒   (𝜑 → 𝐶 ∈ TermCat)
 
19-Oct-2025idfudiag1bas 49649 If the identity functor of a category is the same as a constant functor to the category, then the base is a singleton. (Contributed by Zhi Wang, 19-Oct-2025.)
𝐼 = (idfunc‘𝐶)    &   𝐿 = (𝐶Δfunc𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   𝐾 = ((1st ‘𝐿)‘𝑋)    &   (𝜑 → 𝐼 = 𝐾)    ⇒   (𝜑 → 𝐵 = {𝑋})
 
19-Oct-2025idfudiag1lem 49648 Lemma for idfudiag1bas 49649 and idfudiag1 49650. (Contributed by Zhi Wang, 19-Oct-2025.)
(𝜑 → ( I ↾ 𝐴) = (𝐴 × {𝐵}))    &   (𝜑 → 𝐴 ≠ ∅)    ⇒   (𝜑 → 𝐴 = {𝐵})
 
19-Oct-2025eufunc 49647 If there exists a unique functor from a non-empty category, then the base of the target category is a singleton. (Contributed by Zhi Wang, 19-Oct-2025.)
(𝜑 → ∃!𝑓 𝑓 ∈ (𝐶 Func 𝐷))    &   𝐴 = (Base‘𝐶)    &   (𝜑 → 𝐴 ≠ ∅)    &   𝐵 = (Base‘𝐷)    ⇒   (𝜑 → ∃!𝑥 𝑥 ∈ 𝐵)
 
19-Oct-2025eufunclem 49646 If there exists a unique functor from a non-empty category, then the base of the target category is at most a singleton. (Contributed by Zhi Wang, 19-Oct-2025.)
(𝜑 → ∃!𝑓 𝑓 ∈ (𝐶 Func 𝐷))    &   𝐴 = (Base‘𝐶)    &   (𝜑 → 𝐴 ≠ ∅)    &   𝐵 = (Base‘𝐷)    ⇒   (𝜑 → 𝐵 ≼ 1o)
 
19-Oct-2025diag1f1 49432 The object part of the diagonal functor is 1-1 if 𝐵 is non-empty. (Contributed by Zhi Wang, 19-Oct-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    &   𝐴 = (Base‘𝐶)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝐵 ≠ ∅)    ⇒   (𝜑 → (1st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶))
 
19-Oct-2025diag1f1lem 49431 The object part of the diagonal functor is 1-1 if 𝐵 is non-empty. Note that (𝜑 → (𝑀 = 𝑁 ↔ 𝑋 = 𝑌)) also holds because of diag1f1 49432 and f1fveq 7202. (Contributed by Zhi Wang, 19-Oct-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    &   𝐴 = (Base‘𝐶)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝐵 ≠ ∅)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   𝑀 = ((1st ‘𝐿)‘𝑋)    &   𝑁 = ((1st ‘𝐿)‘𝑌)    ⇒   (𝜑 → (𝑀 = 𝑁 → 𝑋 = 𝑌))
 
19-Oct-2025diag1a 49430 The constant functor of 𝑋. (Contributed by Zhi Wang, 19-Oct-2025.)
𝐿 = (𝐶Δfunc𝐷)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    &   𝐴 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   𝐾 = ((1st ‘𝐿)‘𝑋)    &   𝐵 = (Base‘𝐷)    &   𝐽 = (Hom ‘𝐷)    &    1 = (Id‘𝐶)    ⇒   (𝜑 → 𝐾 = ⟨(𝐵 × {𝑋}), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦𝐽𝑧) × {( 1 ‘𝑋)}))⟩)
 
19-Oct-2025func0g2 49215 The source category of a functor to the empty category must be empty as well. (Contributed by Zhi Wang, 19-Oct-2025.)
𝐴 = (Base‘𝐶)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝐵 = ∅)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → 𝐴 = ∅)
 
19-Oct-2025func0g 49214 The source category of a functor to the empty category must be empty as well. (Contributed by Zhi Wang, 19-Oct-2025.)
𝐴 = (Base‘𝐶)    &   𝐵 = (Base‘𝐷)    &   (𝜑 → 𝐵 = ∅)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   (𝜑 → 𝐴 = ∅)
 
19-Oct-2025func1st2nd 49201 Rewrite the functor predicate with separated parts. (Contributed by Zhi Wang, 19-Oct-2025.)
(𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
 
19-Oct-2025wfac8prim 45119 The class of well-founded sets 𝑊 models the Axiom of Choice. Since the previous theorems show that all the ZF axioms hold in 𝑊, we may use any statement that ZF proves is equivalent to Choice to prove this. We use ac8prim 45108. Part of Corollary II.2.12 of [Kunen2] p. 114. (Contributed by Eric Schmidt, 19-Oct-2025.)
𝑊 = ∪ (𝑅1 “ On)    ⇒   ∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑊 ∀𝑤 ∈ 𝑊 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑊 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 ∀𝑣 ∈ 𝑊 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))
 
19-Oct-2025wfaxinf2 45118 The class of well-founded sets models the Axiom of Infinity ax-inf2 9538. Part of Corollary II.2.12 of [Kunen2] p. 114. (Contributed by Eric Schmidt, 19-Oct-2025.)
𝑊 = ∪ (𝑅1 “ On)    ⇒   ∃𝑥 ∈ 𝑊 (∃𝑦 ∈ 𝑊 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑊 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑊 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
 
19-Oct-2025wfaxreg 45117 The class of well-founded sets models the Axiom of Regularity ax-reg 9485. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 19-Oct-2025.)
𝑊 = ∪ (𝑅1 “ On)    ⇒   ∀𝑥 ∈ 𝑊 (∃𝑦 ∈ 𝑊 𝑦 ∈ 𝑥 → ∃𝑦 ∈ 𝑊 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
 
19-Oct-2025wfaxun 45116 The class of well-founded sets models the Axiom of Union ax-un 7674. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 19-Oct-2025.)
𝑊 = ∪ (𝑅1 “ On)    ⇒   ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∃𝑤 ∈ 𝑊 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
 
19-Oct-2025wfaxpow 45114 The class of well-founded sets models the Axioms of Power Sets. Part of Corollary II.2.9 of [Kunen2] p. 113. (Contributed by Eric Schmidt, 19-Oct-2025.)
𝑊 = ∪ (𝑅1 “ On)    ⇒   ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
 
19-Oct-2025wfaxnul 45113 The class of well-founded sets models the Null Set Axiom ax-nul 5246. (Contributed by Eric Schmidt, 19-Oct-2025.)
𝑊 = ∪ (𝑅1 “ On)    ⇒   ∃𝑥 ∈ 𝑊 ∀𝑦 ∈ 𝑊 ¬ 𝑦 ∈ 𝑥
 
19-Oct-2025modelac8prim 45109 If 𝑀 is a transitive class, then the following are equivalent. (1) Every nonempty set 𝑥 ∈ 𝑀 of pairwise disjoint nonempty sets has a choice set in 𝑀. (2) The class 𝑀 models the Axiom of Choice, in the form ac8prim 45108.

Lemma II.2.11(7) of [Kunen2] p. 114. Kunen has the additional hypotheses that the Extensionality, Separation, Pairing, and Union axioms are true in 𝑀. This, apparently, is because Kunen's statement of the Axiom of Choice uses defined notions, including ∅ and ∩, and these axioms guarantee that these notions are well-defined. When we state the axiom using primitives only, the need for these hypotheses disappears. (Contributed by Eric Schmidt, 19-Oct-2025.)

(Tr 𝑀 → (∀𝑥 ∈ 𝑀 ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝑀 ((∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))))
 
19-Oct-2025ac8prim 45108 ac8 10390 expanded into primitives. (Contributed by Eric Schmidt, 19-Oct-2025.)
((∀𝑧(𝑧 ∈ 𝑥 → ∃𝑤 𝑤 ∈ 𝑧) ∧ ∀𝑧∀𝑤((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑥 → ∃𝑤∀𝑣((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))
 
19-Oct-2025dfac5prim 45107 dfac5 10027 expanded into primitives. (Contributed by Eric Schmidt, 19-Oct-2025.)
(CHOICE ↔ ∀𝑥((∀𝑧(𝑧 ∈ 𝑥 → ∃𝑤 𝑤 ∈ 𝑧) ∧ ∀𝑧∀𝑤((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑥 → ∃𝑤∀𝑣((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤))))

Older news:

(29-Jul-2020) Mario Carneiro presented MM0 at the CICM conference. See this Google Group post which includes a YouTube link.

(20-Jul-2020) Rohan Ridenour found 5 shorter D-proofs in our Shortest known proofs... file. In particular, he reduced *4.39 from 901 to 609 steps. A note on the Metamath Solitaire page mentions a tool that he worked with.

(19-Jul-2020) David A. Wheeler posted a video (https://youtu.be/3R27Qx69jHc) on how to (re)prove Schwabh�user 4.6 for the Metamath Proof Explorer. See also his older videos.

(19-Jul-2020) In version 0.184 of the metamath program, "verify markup" now checks that mathboxes are independent i.e. do not cross-reference each other. To turn off this check, use "/mathbox_skip"

(30-Jun-2020) In version 0.183 of the metamath program, (1) "verify markup" now has checking for (i) underscores in labels, (ii) that *ALT and *OLD theorems have both discouragement tags, and (iii) that lines don't have trailing spaces. (2) "save proof.../rewrap" no longer left-aligns $p/$a comments that contain the string "<HTML>"; see this note.

(5-Apr-2020) Glauco Siliprandi added a new proof to the 100 theorem list, e is Transcendental etransc, bringing the Metamath total to 74.

(12-Feb-2020) A bug in the 'minimize' command of metamath.exe versions 0.179 (29-Nov-2019) and 0.180 (10-Dec-2019) may incorrectly bring in the use of new axioms. Version 0.181 fixes it.

(20-Jan-2020) David A. Wheeler created a video called Walkthrough of the tutorial in mmj2. See the Google Group announcement for more details. (All of his videos are listed on the Other Metamath-Related Topics page.)

(18-Jan-2020) The FOMM 2020 talks are on youtube now. Mario Carneiro's talk is Metamath Zero, or: How to Verify a Verifier. Since they are washed out in the video, the PDF slides are available separately.

(14-Dec-2019) Glauco Siliprandi added a new proof to the 100 theorem list, Fourier series convergence fourier, bringing the Metamath total to 73.

(25-Nov-2019) Alexander van der Vekens added a new proof to the 100 theorem list, The Cayley-Hamilton Theorem cayleyhamilton, bringing the Metamath total to 72.

(25-Oct-2019) Mario Carneiro's paper "Metamath Zero: The Cartesian Theorem Prover" (submitted to CPP 2020) is now available on arXiv: https://arxiv.org/abs/1910.10703. There is a related discussion on Hacker News.

(30-Sep-2019) Mario Carneiro's talk about MM0 at ITP 2019 is available on YouTube: x86 verification from scratch (24 minutes). Google Group discussion: Metamath Zero.

(29-Sep-2019) David Wheeler created a fascinating Gource video that animates the construction of set.mm, available on YouTube: Metamath set.mm contributions viewed with Gource through 2019-09-26 (4 minutes). Google Group discussion: Gource video of set.mm contributions.

(24-Sep-2019) nLab added a page for Metamath. It mentions Stefan O'Rear's Busy Beaver work using the set.mm axiomatization (and fails to mention Mario's definitional soundness checker)

(1-Sep-2019) Xuanji Li published a Visual Studio Code extension to support metamath syntax highlighting.

(10-Aug-2019) (revised 21-Sep-2019) Version 0.178 of the metamath program has the following changes: (1) "minimize_with" will now prevent dependence on new $a statements unless the new qualifier "/allow_new_axioms" is specified. For routine usage, it is suggested that you use "minimize_with * /allow_new_axioms * /no_new_axioms_from ax-*" instead of just "minimize_with *". See "help minimize_with" and this Google Group post. Also note that the qualifier "/allow_growth" has been renamed to "/may_grow". (2) "/no_versioning" was added to "write theorem_list".

(8-Jul-2019) Jon Pennant announced the creation of a Metamath search engine. Try it and feel free to comment on it at https://groups.google.com/d/msg/metamath/cTeU5AzUksI/5GesBfDaCwAJ.

(16-May-2019) Set.mm now has a major new section on elementary geometry. This begins with definitions that implement Tarski's axioms of geometry (including concepts such as congruence and betweenness). This uses set.mm's extensible structures, making them easier to use for many circumstances. The section then connects Tarski geometry with geometry in Euclidean places. Most of the work in this section is due to Thierry Arnoux, with earlier work by Mario Carneiro and Scott Fenton. [Reported by DAW.]

(9-May-2019) We are sad to report that long-time contributor Alan Sare passed away on Mar. 23. There is some more information at the top of his mathbox (click on "Mathbox for Alan Sare") and his obituary. We extend our condolences to his family.

(10-Mar-2019) Jon Pennant and Mario Carneiro added a new proof to the 100 theorem list, Heron's formula heron, bringing the Metamath total to 71.

(22-Feb-2019) Alexander van der Vekens added a new proof to the 100 theorem list, Cramer's rule cramer, bringing the Metamath total to 70.

(6-Feb-2019) David A. Wheeler has made significant improvements and updates to the Metamath book. Any comments, errors found, or suggestions are welcome and should be turned into an issue or pull request at https://github.com/metamath/metamath-book (or sent to me if you prefer).

(26-Dec-2018) I added Appendix 8 to the MPE Home Page that cross-references new and old axiom numbers.

(20-Dec-2018) The axioms have been renumbered according to this Google Groups post.

(24-Nov-2018) Thierry Arnoux created a new page on topological structures. The page along with its SVG files are maintained on GitHub.

(11-Oct-2018) Alexander van der Vekens added a new proof to the 100 theorem list, the Friendship Theorem friendship, bringing the Metamath total to 69.

(1-Oct-2018) Naip Moro has written gramm, a Metamath proof verifier written in Antlr4/Java.

(16-Sep-2018) The definition df-riota has been simplified so that it evaluates to the empty set instead of an Undef value. This change affects a significant part of set.mm.

(2-Sep-2018) Thierry Arnoux added a new proof to the 100 theorem list, Euler's partition theorem eulerpart, bringing the Metamath total to 68.

(1-Sep-2018) The Kate editor now has Metamath syntax highlighting built in. (Communicated by Wolf Lammen.)

(15-Aug-2018) The Intuitionistic Logic Explorer now has a Most Recent Proofs page.

(4-Aug-2018) Version 0.163 of the metamath program now indicates (with an asterisk) which Table of Contents headers have associated comments.

(10-May-2018) George Szpiro, journalist and author of several books on popular mathematics such as Poincare's Prize and Numbers Rule, used a genetic algorithm to find shorter D-proofs of "*3.37" and "meredith" in our Shortest known proofs... file.

(19-Apr-2018) The EMetamath Eclipse plugin has undergone many improvements since its initial release as the change log indicates. Thierry uses it as his main proof assistant and writes, "I added support for mmj2's auto-transformations, which allows it to infer several steps when building proofs. This added a lot of comfort for writing proofs.... I can now switch back and forth between the proof assistant and editing the Metamath file.... I think no other proof assistant has this feature."

(11-Apr-2018) Benoît Jubin solved an open problem about the "Axiom of Twoness," showing that it is necessary for completeness. See item 14 on the "Open problems and miscellany" page.

(25-Mar-2018) Giovanni Mascellani has announced mmpp, a new proof editing environment for the Metamath language.

(27-Feb-2018) Bill Hale has released an app for the Apple iPad and desktop computer that allows you to browse Metamath theorems and their proofs.

(17-Jan-2018) Dylan Houlihan has kindly provided a new mirror site. He has also provided an rsync server; type "rsync uk.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").

(15-Jan-2018) The metamath program, version 0.157, has been updated to implement the file inclusion conventions described in the 21-Dec-2017 entry of mmnotes.txt.

(11-Dec-2017) I added a paragraph, suggested by Gérard Lang, to the distinct variable description here.

(10-Dec-2017) Per FL's request, his mathbox will be removed from set.mm. If you wish to export any of his theorems, today's version (master commit 1024a3a) is the last one that will contain it.

(11-Nov-2017) Alan Sare updated his completeusersproof program.

(3-Oct-2017) Sean B. Palmer created a web page that runs the metamath program under emulated Linux in JavaScript. He also wrote some programs to work with our shortest known proofs of the PM propositional calculus theorems.

(28-Sep-2017) Ivan Kuckir wrote a tutorial blog entry, Introduction to Metamath, that summarizes the language syntax. (It may have been written some time ago, but I was not aware of it before.)

(26-Sep-2017) The default directory for the Metamath Proof Explorer (MPE) has been changed from the GIF version (mpegif) to the Unicode version (mpeuni) throughout the site. Please let me know if you find broken links or other issues.

(24-Sep-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Ceva's Theorem cevath, bringing the Metamath total to 67.

(3-Sep-2017) Brendan Leahy added a new proof to the 100 theorem list, Area of a Circle areacirc, bringing the Metamath total to 66.

(7-Aug-2017) Mario Carneiro added a new proof to the 100 theorem list, Principle of Inclusion/Exclusion incexc, bringing the Metamath total to 65.

(1-Jul-2017) Glauco Siliprandi added a new proof to the 100 theorem list, Stirling's Formula stirling, bringing the Metamath total to 64. Related theorems include 2 versions of Wallis' formula for π (wallispi and wallispi2).

(7-May-2017) Thierry Arnoux added a new proof to the 100 theorem list, Betrand's Ballot Problem ballotth, bringing the Metamath total to 63.

(20-Apr-2017) Glauco Siliprandi added a new proof in the supplementary list on the 100 theorem list, Stone-Weierstrass Theorem stowei.

(28-Feb-2017) David Moews added a new proof to the 100 theorem list, Product of Segments of Chords chordthm, bringing the Metamath total to 62.

(1-Jan-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Isosceles triangle theorem isosctr, bringing the Metamath total to 61.

(1-Jan-2017) Mario Carneiro added 2 new proofs to the 100 theorem list, L'Hôpital's Rule lhop and Taylor's Theorem taylth, bringing the Metamath total to 60.

(28-Dec-2016) David A. Wheeler is putting together a page on Metamath (specifically set.mm) conventions. Comments are welcome on the Google Group thread.

(24-Dec-2016) Mario Carneiro introduced the abbreviation "F/ x ph" (symbols: turned F, x, phi) in df-nf to represent the "effectively not free" idiom "A. x ( ph -> A. x ph )". Theorem nf2 shows a version without nested quantifiers.

(22-Dec-2016) Naip Moro has developed a Metamath database for G. Spencer-Brown's Laws of Form. You can follow the Google Group discussion here.

(20-Dec-2016) In metamath program version 0.137, 'verify markup *' now checks that ax-XXX $a matches axXXX $p when the latter exists, per the discussion at https://groups.google.com/d/msg/metamath/Vtz3CKGmXnI/Fxq3j1I_EQAJ.

(24-Nov-2016) Mingl Yuan has kindly provided a mirror site in Beijing, China. He has also provided an rsync server; type "rsync cn.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").

(14-Aug-2016) All HTML pages on this site should now be mobile-friendly and pass the Mobile-Friendly Test. If you find one that does not, let me know.

(14-Aug-2016) Daniel Whalen wrote a paper describing the use of using deep learning to prove 14% of test theorems taken from set.mm: Holophrasm: a neural Automated Theorem Prover for higher-order logic. The associated program is called Holophrasm.

(14-Aug-2016) David A. Wheeler created a video called Metamath Proof Explorer: A Modern Principia Mathematica

(12-Aug-2016) A Gitter chat room has been created for Metamath.

(9-Aug-2016) Mario Carneiro wrote a Metamath proof verifier in the Scala language as part of the ongoing Metamath -> MMT import project

(9-Aug-2016) David A. Wheeler created a GitHub project called metamath-test (last execution run) to check that different verifiers both pass good databases and detect errors in defective ones.

(4-Aug-2016) Mario gave two presentations at CICM 2016.

(17-Jul-2016) Thierry Arnoux has written EMetamath, a Metamath plugin for the Eclipse IDE.

(16-Jul-2016) Mario recovered Chris Capel's collapsible proof demo.

(13-Jul-2016) FL sent me an updated version of PDF (LaTeX source) developed with Lamport's pf2 package. See the 23-Apr-2012 entry below.

(12-Jul-2016) David A. Wheeler produced a new video for mmj2 called "Creating functions in Metamath". It shows a more efficient approach than his previous recent video "Creating functions in Metamath" (old) but it can be of interest to see both approaches.

(10-Jul-2016) Metamath program version 0.132 changes the command 'show restricted' to 'show discouraged' and adds a new command, 'set discouragement'. See the mmnotes.txt entry of 11-May-2016 (updated 10-Jul-2016).

(12-Jun-2016) Dan Getz has written Metamath.jl, a Metamath proof verifier written in the Julia language.

(10-Jun-2016) If you are using metamath program versions 0.128, 0.129, or 0.130, please update to version 0.131. (In the bad versions, 'minimize_with' ignores distinct variable violations.)

(1-Jun-2016) Mario Carneiro added new proofs to the 100 theorem list, the Prime Number Theorem pnt and the Perfect Number Theorem perfect, bringing the Metamath total to 58.

(12-May-2016) Mario Carneiro added a new proof to the 100 theorem list, Dirichlet's theorem dirith, bringing the Metamath total to 56. (Added 17-May-2016) An informal exposition of the proof can be found at http://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html

(10-Mar-2016) Metamath program version 0.125 adds a new qualifier, /fast, to 'save proof'. See the mmnotes.txt entry of 10-Mar-2016.

(6-Mar-2016) The most recent set.mm has a large update converting variables from letters to symbols. See this Google Groups post.

(16-Feb-2016) Mario Carneiro's new paper "Models for Metamath" can be found here and on arxiv.org.

(6-Feb-2016) There are now 22 math symbols that can be used as variable names. See mmascii.html near the 50th table row, starting with "./\".

(29-Jan-2016) Metamath program version 0.123 adds /packed and /explicit qualifiers to 'save proof' and 'show proof'. See this Google Groups post.

(13-Jan-2016) The Unicode math symbols now provide for external CSS and use the XITS web font. Thanks to David A. Wheeler, Mario Carneiro, Cris Perdue, Jason Orendorff, and Frédéric Liné for discussions on this topic. Two commands, htmlcss and htmlfont, were added to the $t comment in set.mm and are recognized by Metamath program version 0.122.

(21-Dec-2015) Axiom ax-12, now renamed ax-12o, was replaced by a new shorter equivalent, ax-12. The equivalence is provided by theorems ax12o and ax12.

(13-Dec-2015) A new section on the theory of classes was added to the MPE Home Page. Thanks to Gérard Lang for suggesting this section and improvements to it.

(17-Nov-2015) Metamath program version 0.121: 'verify markup' was added to check comment markup consistency; see 'help verify markup'. You are encouraged to make sure 'verify markup */f' has no warnings prior to mathbox submissions. The date consistency rules are given in this Google Groups post.

(23-Sep-2015) Drahflow wrote, "I am currently working on yet another proof assistant, main reason being: I understand stuff best if I code it. If anyone is interested: https://github.com/Drahflow/Igor (but in my own programming language, so expect a complicated build process :P)"

(23-Aug-2015) Ivan Kuckir created MM Tool, a Metamath proof verifier and editor written in JavaScript that runs in a browser.

(25-Jul-2015) Axiom ax-10 is shown to be redundant by theorem ax10 , so it was removed from the predicate calculus axiom list.

(19-Jul-2015) Mario Carneiro gave two talks related to Metamath at CICM 2015, which are linked to at Other Metamath-Related Topics.

(18-Jul-2015) The metamath program has been updated to version 0.118. 'show trace_back' now has a '/to' qualifier to show the path back to a specific axiom such as ax-ac. See 'help show trace_back'.

(12-Jul-2015) I added the HOL Explorer for Mario Carneiro's hol.mm database. Although the home page needs to be filled out, the proofs can be accessed.

(11-Jul-2015) I started a new page, Other Metamath-Related Topics, that will hold miscellaneous material that doesn't fit well elsewhere (or is hard to find on this site). Suggestions welcome.

(23-Jun-2015) Metamath's mascot, Penny the cat (2007 photo), passed away today. She was 18 years old.

(21-Jun-2015) Mario Carneiro added 3 new proofs to the 100 theorem list: All Primes (1 mod 4) Equal the Sum of Two Squares 2sq, The Law of Quadratic Reciprocity lgsquad and the AM-GM theorem amgm, bringing the Metamath total to 55.

(13-Jun-2015) Stefan O'Rear's smm, written in JavaScript, can now be used as a standalone proof verifier. This brings the total number of independent Metamath verifiers to 8, written in just as many languages (C, Java. JavaScript, Python, Haskell, Lua, C#, C++).

(12-Jun-2015) David A. Wheeler added 2 new proofs to the 100 theorem list: The Law of Cosines lawcos and Ptolemy's Theorem ptolemy, bringing the Metamath total to 52.

(30-May-2015) The metamath program has been updated to version 0.117. (1) David A. Wheeler provided an enhancement to speed up the 'improve' command by 28%; see README.TXT for more information. (2) In web pages with proofs, local hyperlinks on step hypotheses no longer clip the Expression cell at the top of the page.

(9-May-2015) Stefan O'Rear has created an archive of older set.mm releases back to 1998: https://github.com/sorear/set.mm-history/.

(7-May-2015) The set.mm dated 7-May-2015 is a major revision, updated by Mario, that incorporates the new ordered pair definition df-op that was agreed upon. There were 700 changes, listed at the top of set.mm. Mathbox users are advised to update their local mathboxes. As usual, if any mathbox user has trouble incorporating these changes into their mathbox in progress, Mario or I will be glad to do them for you.

(7-May-2015) Mario has added 4 new theorems to the 100 theorem list: Ramsey's Theorem ramsey, The Solution of a Cubic cubic, The Solution of the General Quartic Equation quart, and The Birthday Problem birthday. In the Supplementary List, Stefan O'Rear added the Hilbert Basis Theorem hbt.

(28-Apr-2015) A while ago, Mario Carneiro wrote up a proof of the unambiguity of set.mm's grammar, which has now been added to this site: grammar-ambiguity.txt.

(22-Apr-2015) The metamath program has been updated to version 0.114. In MM-PA, 'show new_proof/unknown' now shows the relative offset (-1, -2,...) used for 'assign' arguments, suggested by Stefan O'Rear.

(20-Apr-2015) I retrieved an old version of the missing "Metamath 100" page from archive.org and updated it to what I think is the current state: mm_100.html. Anyone who wants to edit it can email updates to this page to me.

(19-Apr-2015) The metamath program has been updated to version 0.113, mostly with patches provided by Stefan O'Rear. (1) 'show statement %' (or any command allowing label wildcards) will select statements whose proofs were changed in current session. ('help search' will show all wildcard matching rules.) (2) 'show statement =' will select the statement being proved in MM-PA. (3) The proof date stamp is now created only if the proof is complete.

(18-Apr-2015) There is now a section for Scott Fenton's NF database: New Foundations Explorer.

(16-Apr-2015) Mario describes his recent additions to set.mm at https://groups.google.com/forum/#!topic/metamath/VAGNmzFkHCs. It include 2 new additions to the Formalizing 100 Theorems list, Leibniz' series for pi (leibpi) and the Konigsberg Bridge problem (konigsberg)

(10-Mar-2015) Mario Carneiro has written a paper, "Arithmetic in Metamath, Case Study: Bertrand's Postulate," for CICM 2015. A preprint is available at arXiv:1503.02349.

(23-Feb-2015) Scott Fenton has created a Metamath formalization of NF set theory: https://github.com/sctfn/metamath-nf/. For more information, see the Metamath Google Group posting.

(28-Jan-2015) Mario Carneiro added Wilson's Theorem (wilth), Ascending or Descending Sequences (erdsze, erdsze2), and Derangements Formula (derangfmla, subfaclim), bringing the Metamath total for Formalizing 100 Theorems to 44.

(19-Jan-2015) Mario Carneiro added Sylow's Theorem (sylow1, sylow2, sylow2b, sylow3), bringing the Metamath total for Formalizing 100 Theorems to 41.

(9-Jan-2015) The hypothesis order of mpbi*an* was changed. See the Notes entry of 9-Jan-2015.

(1-Jan-2015) Mario Carneiro has written a paper, "Conversion of HOL Light proofs into Metamath," that has been submitted to the Journal of Formalized Reasoning. A preprint is available on arxiv.org.

(22-Nov-2014) Stefan O'Rear added the Solutions to Pell's Equation (rmxycomplete) and Liouville's Theorem and the Construction of Transcendental Numbers (aaliou), bringing the Metamath total for Formalizing 100 Theorems to 40.

(22-Nov-2014) The metamath program has been updated with version 0.111. (1) Label wildcards now have a label range indicator "~" so that e.g. you can show or search all of the statements in a mathbox. See 'help search'. (Stefan O'Rear added this to the program.) (2) A qualifier was added to 'minimize_with' to prevent the use of any axioms not already used in the proof e.g. 'minimize_with * /no_new_axioms_from ax-*' will prevent the use of ax-ac if the proof doesn't already use it. See 'help minimize_with'.

(10-Oct-2014) Mario Carneiro has encoded the axiomatic basis for the HOL theorem prover into a Metamath source file, hol.mm.

(24-Sep-2014) Mario Carneiro added the Sum of the Angles of a Triangle (ang180), bringing the Metamath total for Formalizing 100 Theorems to 38.

(15-Sep-2014) Mario Carneiro added the Fundamental Theorem of Algebra (fta), bringing the Metamath total for Formalizing 100 Theorems to 37.

(3-Sep-2014) Mario Carneiro added the Fundamental Theorem of Integral Calculus (ftc1, ftc2). This brings the Metamath total for Formalizing 100 Theorems to 35. (added 14-Sep-2014) Along the way, he added the Mean Value Theorem (mvth), bringing the total to 36.

(16-Aug-2014) Mario Carneiro started a Metamath blog at http://metamath-blog.blogspot.com/.

(10-Aug-2014) Mario Carneiro added Erdős's proof of the divergence of the inverse prime series (prmrec). This brings the Metamath total for Formalizing 100 Theorems to 34.

(31-Jul-2014) Mario Carneiro added proofs for Euler's Summation of 1 + (1/2)^2 + (1/3)^2 + .... (basel) and The Factor and Remainder Theorems (facth, plyrem). This brings the Metamath total for Formalizing 100 Theorems to 33.

(16-Jul-2014) Mario Carneiro added proofs for Four Squares Theorem (4sq), Formula for the Number of Combinations (hashbc), and Divisibility by 3 Rule (3dvds). This brings the Metamath total for Formalizing 100 Theorems to 31.

(11-Jul-2014) Mario Carneiro added proofs for Divergence of the Harmonic Series (harmonic), Order of a Subgroup (lagsubg), and Lebesgue Measure and Integration (itgcl). This brings the Metamath total for Formalizing 100 Theorems to 28.

(7-Jul-2014) Mario Carneiro presented a talk, "Natural Deduction in the Metamath Proof Language," at the 6PCM conference. Slides Audio

(25-Jun-2014) In version 0.108 of the metamath program, the 'minimize_with' command is now more automated. It now considers compressed proof length; it scans the statements in forward and reverse order and chooses the best; and it avoids $d conflicts. The '/no_distinct', '/brief', and '/reverse' qualifiers are obsolete, and '/verbose' no longer lists all statements scanned but gives more details about decision criteria.

(12-Jun-2014) To improve naming uniformity, theorems about operation values now use the abbreviation "ov". For example, df-opr, opreq1, oprabval5, and oprvres are now called df-ov, oveq1, ov5, and ovres respectively.

(11-Jun-2014) Mario Carneiro finished a major revision of set.mm. His notes are under the 11-Jun-2014 entry in the Notes

(4-Jun-2014) Mario Carneiro provided instructions and screenshots for syntax highlighting for the jEdit editor for use with Metamath and mmj2 source files.

(19-May-2014) Mario Carneiro added a feature to mmj2, in the build at https://github.com/digama0/mmj2/raw/dev-build/mmj2jar/mmj2.jar, which tests all but 5 definitions in set.mm for soundness. You can turn on the test by adding
SetMMDefinitionsCheckWithExclusions,ax-*,df-bi,df-clab,df-cleq,df-clel,df-sbc
to your RunParms.txt file.

(17-May-2014) A number of labels were changed in set.mm, listed at the top of set.mm as usual. Note in particular that the heavily-used visset, elisseti, syl11anc, syl111anc were changed respectively to vex, elexi, syl2anc, syl3anc.

(16-May-2014) Scott Fenton formalized a proof for "Sum of kth powers": fsumkthpow. This brings the Metamath total for Formalizing 100 Theorems to 25.

(9-May-2014) I (Norm Megill) presented an overview of Metamath at the "Formalization of mathematics in proof assistants" workshop at the Institut Henri Poincar� in Paris. The slides for this talk are here.

(22-Jun-2014) Version 0.107 of the metamath program adds a "PART" indention level to the Statement List table of contents, adds 'show proof ... /size' to show source file bytes used, and adds 'show elapsed_time'. The last one is helpful for measuring the run time of long commands. See 'help write theorem_list', 'help show proof', and 'help show elapsed_time' for more information.

(2-May-2014) Scott Fenton formalized a proof of Sum of the Reciprocals of the Triangular Numbers: trirecip. This brings the Metamath total for Formalizing 100 Theorems to 24.

(19-Apr-2014) Scott Fenton formalized a proof of the Formula for Pythagorean Triples: pythagtrip. This brings the Metamath total for Formalizing 100 Theorems to 23.

(11-Apr-2014) David A. Wheeler produced a much-needed and well-done video for mmj2, called "Introduction to Metamath & mmj2". Thanks, David!

(15-Mar-2014) Mario Carneiro formalized a proof of Bertrand's postulate: bpos. This brings the Metamath total for Formalizing 100 Theorems to 22.

(18-Feb-2014) Mario Carneiro proved that complex number axiom ax-cnex is redundant (theorem cnex). See also Real and Complex Numbers.

(11-Feb-2014) David A. Wheeler has created a theorem compilation that tracks those theorems in Freek Wiedijk's Formalizing 100 Theorems list that have been proved in set.mm. If you find a error or omission in this list, let me know so it can be corrected. (Update 1-Mar-2014: Mario has added eulerth and bezout to the list.)

(4-Feb-2014) Mario Carneiro writes:

The latest commit on the mmj2 development branch introduced an exciting new feature, namely syntax highlighting for mmp files in the main window. (You can pick up the latest mmj2.jar at https://github.com/digama0/mmj2/blob/develop/mmj2jar/mmj2.jar .) The reason I am asking for your help at this stage is to help with design for the syntax tokenizer, which is responsible for breaking down the input into various tokens with names like "comment", "set", and "stephypref", which are then colored according to the user's preference. As users of mmj2 and metamath, what types of highlighting would be useful to you?

One limitation of the tokenizer is that since (for performance reasons) it can be started at any line in the file, highly contextual coloring, like highlighting step references that don't exist previously in the file, is difficult to do. Similarly, true parsing of the formulas using the grammar is possible but likely to be unmanageably slow. But things like checking theorem labels against the database is quite simple to do under the current setup.

That said, how can this new feature be optimized to help you when writing proofs?

(13-Jan-2014) Mathbox users: the *19.21a*, *19.23a* series of theorems have been renamed to *alrim*, *exlim*. You can update your mathbox with a global replacement of string '19.21a' with 'alrim' and '19.23a' with 'exlim'.

(5-Jan-2014) If you downloaded mmj2 in the past 3 days, please update it with the current version, which fixes a bug introduced by the recent changes that made it unable to read in most of the proofs in the textarea properly.

(4-Jan-2014) I added a list of "Allowed substitutions" under the "Distinct variable groups" list on the theorem web pages, for example axsep. This is an experimental feature and comments are welcome.

(3-Jan-2014) Version 0.102 of the metamath program produces more space-efficient compressed proofs (still compatible with the specification in Appendix B of the Metamath book) using an algorithm suggested by Mario Carneiro. See 'help save proof' in the program. Also, mmj2 now generates proofs in the new format. The new mmj2 also has a mandatory update that fixes a bug related to the new format; you must update your mmj2 copy to use it with the latest set.mm.

(23-Dec-2013) Mario Carneiro has updated many older definitions to use the maps-to notation. If you have difficulty updating your local mathbox, contact him or me for assistance.

(1-Nov-2013) 'undo' and 'redo' commands were added to the Proof Assistant in metamath program version 0.07.99. See 'help undo' in the program.

(8-Oct-2013) Today's Notes entry describes some proof repair techniques.

(5-Oct-2013) Today's Notes entry explains some recent extensible structure improvements.

(8-Sep-2013) Mario Carneiro has revised the square root and sequence generator definitions. See today's Notes entry.

(3-Aug-2013) Mario Carneiro writes: "I finally found enough time to create a GitHub repository for development at https://github.com/digama0/mmj2. A permalink to the latest version plus source (akin to mmj2.zip) is https://github.com/digama0/mmj2/zipball/, and the jar file on its own (mmj2.jar) is at https://github.com/digama0/mmj2/blob/master/mmj2jar/mmj2.jar?raw=true. Unfortunately there is no easy way to automatically generate mmj2jar.zip, but this is available as part of the zip distribution for mmj2.zip. History tracking will be handled by the repository now. Do you have old versions of the mmj2 directory? I could add them as historical commits if you do."

(18-Jun-2013) Mario Carneiro has done a major revision and cleanup of the construction of real and complex numbers. In particular, rather than using equivalence classes as is customary for the construction of the temporary rationals, he used only "reduced fractions", so that the use of the axiom of infinity is avoided until it becomes necessary for the construction of the temporary reals.

(18-May-2013) Mario Carneiro has added the ability to produce compressed proofs to mmj2. This is not an official release but can be downloaded here if you want to try it: mmj2.jar. If you have any feedback, send it to me (NM), and I will forward it to Mario. (Disclaimer: this release has not been endorsed by Mel O'Cat. If anyone has been in contact with him, please let me know.)

(29-Mar-2013) Charles Greathouse reduced the size of our PNG symbol images using the pngout program.

(8-Mar-2013) Wolf Lammen has reorganized the theorems in the "Logical negation" section of set.mm into a more orderly, less scattered arrangement.

(27-Feb-2013) Scott Fenton has done a large cleanup of set.mm, eliminating *OLD references in 144 proofs. See the Notes entry for 27-Feb-2013.

(21-Feb-2013) *ATTENTION MATHBOX USERS* The order of hypotheses of many syl* theorems were changed, per a suggestion of Mario Carneiro. You need to update your local mathbox copy for compatibility with the new set.mm, or I can do it for you if you wish. See the Notes entry for 21-Feb-2013.

(16-Feb-2013) Scott Fenton shortened the direct-from-axiom proofs of *3.1, *3.43, *4.4, *4.41, *4.5, *4.76, *4.83, *5.33, *5.35, *5.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).

(27-Jan-2013) Scott Fenton writes, "I've updated Ralph Levien's mmverify.py. It's now a Python 3 program, and supports compressed proofs and file inclusion statements. This adds about fifty lines to the original program. Enjoy!"

(10-Jan-2013) A new mathbox was added for Mario Carneiro, who has contributed a number of cardinality theorems without invoking the Axiom of Choice. This is nice work, and I will be using some of these (those suffixed with "NEW") to replace the existing ones in the main part of set.mm that currently invoke AC unnecessarily.

(4-Jan-2013) As mentioned in the 19-Jun-2012 item below, Eric Schmidt discovered that the complex number axioms axaddcom (now addcom) and ax0id (now addid1) are redundant (schmidt-cnaxioms.pdf, .tex). In addition, ax1id (now mulid1) can be weakened to ax1rid. Scott Fenton has now formalized this work, so that now there are 23 instead of 25 axioms for real and complex numbers in set.mm. The Axioms for Complex Numbers page has been updated with these results. An interesting part of the proof, showing how commutativity of addition follows from other laws, is in addcomi.

(27-Nov-2012) The frequently-used theorems "an1s", "an1rs", "ancom13s", "ancom31s" were renamed to "an12s", "an32s", "an13s", "an31s" to conform to the convention for an12 etc.

(4-Nov-2012) The changes proposed in the Notes, renaming Grp to GrpOp etc., have been incorporated into set.mm. See the list of changes at the top of set.mm. If you want me to update your mathbox with these changes, send it to me along with the version of set.mm that it works with.

(20-Sep-2012) Mel O'Cat updated https://us.metamath.org/ocat/mmj2/TESTmmj2jar.zip. See the README.TXT for a description of the new features.

(21-Aug-2012) Mel O'Cat has uploaded SearchOptionsMockup9.zip, a mockup for the new search screen in mmj2. See the README.txt file for instructions. He will welcome feedback via x178g243 at yahoo.com.

(19-Jun-2012) Eric Schmidt has discovered that in our axioms for complex numbers, axaddcom and ax0id are redundant. (At some point these need to be formalized for set.mm.) He has written up these and some other nice results, including some independence results for the axioms, in schmidt-cnaxioms.pdf (schmidt-cnaxioms.tex).

(23-Apr-2012) Frédéric Liné sent me a PDF (LaTeX source) developed with Lamport's pf2 package. He wrote: "I think it works well with Metamath since the proofs are in a tree form. I use it to have a sketch of a proof. I get this way a better understanding of the proof and I can cut down its size. For instance, inpreima5 was reduced by 50% when I wrote the corresponding proof with pf2."

(5-Mar-2012) I added links to Wikiproofs and its recent changes in the "Wikis" list at the top of this page.

(12-Jan-2012) Thanks to William Hoza who sent me a ZFC T-shirt, and thanks to the ZFC models (courtesy of the Inaccessible Cardinals agency).

FrontBackDetail
ZFC T-shirt front ZFC T-shirt back ZFC T-shirt detail

(24-Nov-2011) In metamath program version 0.07.71, the 'minimize_with' command by default now scans from bottom to top instead of top to bottom, since empirically this often (although not always) results in a shorter proof. A top to bottom scan can be specified with a new qualifier '/reverse'. You can try both methods (starting from the same original proof, of course) and pick the shorter proof.

(15-Oct-2011) From Mel O'Cat:
I just uploaded mmj2.zip containing the 1-Nov-2011 (20111101) release: https://us.metamath.org/ocat/mmj2/mmj2.zip https://us.metamath.org/ocat/mmj2/mmj2.md5
A few last minute tweaks:
1. I now bless double-click starting of mmj2.bat (MacMMJ2.command in Mac OS-X)! See mmj2\QuickStart.html
2. Much improved support of Mac OS-X systems. See mmj2\QuickStart.html
3. I tweaked the Command Line Argument Options report to
a) print every time;
b) print as much as possible even if there are errors in the command line arguments -- and the last line printed corresponds to the argument in error;
c) removed Y/N argument on the command line to enable/disable the report. this simplifies things.
4) Documentation revised, including the PATutorial.
See CHGLOG.TXT for list of all changes. Good luck. And thanks for all of your help!

(15-Sep-2011) MATHBOX USERS: I made a large number of label name changes to set.mm to improve naming consistency. There is a script at the top of the current set.mm that you can use to update your mathbox or older set.mm. Or if you wish, I can do the update on your next mathbox submission - in that case, please include a .zip of the set.mm version you used.

(30-Aug-2011) Scott Fenton shortened the direct-from-axiom proofs of *3.33, *3.45, *4.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).

(21-Aug-2011) A post on reddit generated 60,000 hits (and a TOS violation notice from my provider...),

(18-Aug-2011) The Metamath Google Group has a discussion of my canonical conjunctions proposal. Any feedback directly to me (Norm Megill) is also welcome.

(4-Jul-2011) John Baker has provided (metamath_kindle.zip) "a modified version of [the] metamath.tex [Metamath] book source that is formatted for the Kindle. If you compile the document the resulting PDF can be loaded into into a Kindle and easily read." (Update: the PDF file is now included also.)

(3-Jul-2011) Nested 'submit' calls are now allowed, in metamath program version 0.07.68. Thus you can create or modify a command file (script) from within a command file then 'submit' it. While 'submit' cannot pass arguments (nor are there plans to add this feature), you can 'substitute' strings in the 'submit' target file before calling it in order to emulate this.

(28-Jun-2011)The metamath program version 0.07.64 adds the '/include_mathboxes' qualifier to 'minimize_with'; by default, 'minimize_with *' will now skip checking user mathboxes. Since mathboxes should be independent from each other, this will help prevent accidental cross-"contamination". Also, '/rewrap' was added to 'write source' to automatically wrap $a and $p comments so as to conform to the current formatting conventions used in set.mm. This means you no longer have to be concerned about line length < 80 etc.

(19-Jun-2011) ATTENTION MATHBOX USERS: The wff variables et, ze, si, and rh are now global. This change was made primarily to resolve some conflicts between mathboxes, but it will also let you avoid having to constantly redeclare these locally in the future. Unfortunately, this change can affect the $f hypothesis order, which can cause proofs referencing theorems that use these variables to fail. All mathbox proofs currently in set.mm have been corrected for this, and you should refresh your local copy for further development of your mathbox. You can correct your proofs that are not in set.mm as follows. Only the proofs that fail under the current set.mm (using version 0.07.62 or later of the metamath program) need to be modified.

To fix a proof that references earlier theorems using et, ze, si, and rh, do the following (using a hypothetical theorem 'abc' as an example): 'prove abc' (ignore error messages), 'delete floating', 'initialize all', 'unify all/interactive', 'improve all', 'save new_proof/compressed'. If your proof uses dummy variables, these must be reassigned manually.

To fix a proof that uses et, ze, si, and rh as local variables, make sure the proof is saved in 'compressed' format. Then delete the local declarations ($v and $f statements) and follow the same steps above to correct the proof.

I apologize for the inconvenience. If you have trouble fixing your proofs, you can contact me for assistance.

Note: Versions of the metamath program before 0.07.62 did not flag an error when global variables were redeclared locally, as it should have according to the spec. This caused these spec violations to go unnoticed in some older set.mm versions. The new error messages are in fact just informational and can be ignored when working with older set.mm versions.

(7-Jun-2011) The metamath program version 0.07.60 fixes a bug with the 'minimize_with' command found by Andrew Salmon.

(12-May-2010) Andrew Salmon shortened many proofs, shown above. For comparison, I have temporarily kept the old version, which is suffixed with OLD, such as oridmOLD for oridm.

(9-Dec-2010) Eric Schmidt has written a Metamath proof verifier in C++, called checkmm.cpp.

(3-Oct-2010) The following changes were made to the tokens in set.mm. The subset and proper subset symbol changes to C_ and C. were made to prevent defeating the parenthesis matching in Emacs. Other changes were made so that all letters a-z and A-Z are now available for variable names. One-letter constants such as _V, _e, and _i are now shown on the web pages with Roman instead of italic font, to disambiguate italic variable names. The new convention is that a prefix of _ indicates Roman font and a prefix of ~ indicates a script (curly) font. Thanks to Stefan Allan and Frédéric Liné for discussions leading to this change.

OldNewDescription
C. _C binomial coefficient
E _E epsilon relation
e _e Euler's constant
I _I identity relation
i _i imaginary unit
V _V universal class
(_ C_ subset
(. C. proper subset
P~ ~P power class
H~ ~H Hilbert space

(25-Sep-2010) The metamath program (version 0.07.54) now implements the current Metamath spec, so footnote 2 on p. 92 of the Metamath book can be ignored.

(24-Sep-2010) The metamath program (version 0.07.53) fixes bug 2106, reported by Michal Burger.

(14-Sep-2010) The metamath program (version 0.07.52) has a revamped LaTeX output with 'show statement xxx /tex', which produces the combined statement, description, and proof similar to the web page generation. Also, 'show proof xxx /lemmon/renumber' now matches the web page step numbers. ('show proof xxx/renumber' still has the indented form conforming to the actual RPN proof, with slightly different numbering.)

(9-Sep-2010) The metamath program (version 0.07.51) was updated with a modification by Stefan Allan that adds hyperlinks the the Ref column of proofs.

(12-Jun-2010) Scott Fenton contributed a D-proof (directly from axioms) of Meredith's single axiom (see the end of pmproofs.txt). A description of Meredith's axiom can be found in theorem meredith.

(11-Jun-2010) A new Metamath mirror was added in Austria, courtesy of Kinder-Enduro.

(28-Feb-2010) Raph Levien's Ghilbert project now has a new Ghilbert site and a Google Group.

(26-Jan-2010) Dmitri Vlasov writes, "I admire the simplicity and power of the metamath language, but still I see its great disadvantage - the proofs in metamath are completely non-manageable by humans without proof assistants. Therefore I decided to develop another language, which would be a higher-level superstructure language towards metamath, and which will support human-readable/writable proofs directly, without proof assistants. I call this language mdl (acronym for 'mathematics development language')." The latest version of Dmitri's translators from metamath to mdl and back can be downloaded from http://mathdevlanguage.sourceforge.net/. Currently only Linux is supported, but Dmitri says is should not be difficult to port it to other platforms that have a g++ compiler.

(11-Sep-2009) The metamath program (version 0.07.48) has been updated to enforce the whitespace requirement of the current spec.

(10-Sep-2009) Matthew Leitch has written an nice article, "How to write mathematics clearly", that briefly mentions Metamath. Overall it makes some excellent points. (I have written to him about a few things I disagree with.)

(28-May-2009) AsteroidMeta is back on-line. Note the URL change.

(12-May-2009) Charles Greathouse wrote a Greasemonkey script to reformat the axiom list on Metamath web site proof pages. This is a beta version; he will appreciate feedback.

(11-May-2009) Stefan Allan modified the metamath program to add the command "show statement xxx /mnemonics", which produces the output file Mnemosyne.txt for use with the Mnemosyne project. The current Metamath program download incorporates this command. Instructions: Create the file mnemosyne.txt with e.g. "show statement ax-* /mnemonics". In the Mnemosyne program, load the file by choosing File->Import then file format "Q and A on separate lines". Notes: (1) Don't try to load all of set.mm, it will crash the program due to a bug in Mnemosyne. (2) On my computer, the arrows in ax-1 don't display. Stefan reports that they do on his computer. (Both are Windows XP.)

(3-May-2009) Steven Baldasty wrote a Metamath syntax highlighting file for the gedit editor. Screenshot.

(1-May-2009) Users on a gaming forum discuss our 2+2=4 proof. Notable comments include "Ew math!" and "Whoever wrote this has absolutely no life."

(12-Mar-2009) Chris Capel has created a Javascript theorem viewer demo that (1) shows substitutions and (2) allows expanding and collapsing proof steps. You are invited to take a look and give him feedback at his Metablog.

(28-Feb-2009) Chris Capel has written a Metamath proof verifier in C#, available at http://pdf23ds.net/bzr/MathEditor/Verifier/Verifier.cs and weighing in at 550 lines. Also, that same URL without the file on it is a Bazaar repository.

(2-Dec-2008) A new section was added to the Deduction Theorem page, called Logic, Metalogic, Metametalogic, and Metametametalogic.

(24-Aug-2008) (From ocat): The 1-Aug-2008 version of mmj2 is ready (mmj2.zip), size = 1,534,041 bytes. This version contains the Theorem Loader enhancement which provides a "sandboxing" capability for user theorems and dynamic update of new theorems to the Metamath database already loaded in memory by mmj2. Also, the new "mmj2 Service" feature enables calling mmj2 as a subroutine, or having mmj2 call your program, and provides access to the mmj2 data structures and objects loaded in memory (i.e. get started writing those Jython programs!) See also mmj2 on AsteroidMeta.

(23-May-2008) Gérard Lang pointed me to Bob Solovay's note on AC and strongly inaccessible cardinals. One of the eventual goals for set.mm is to prove the Axiom of Choice from Grothendieck's axiom, like Mizar does, and this note may be helpful for anyone wanting to attempt that. Separately, I also came across a history of the size reduction of grothprim (viewable in Firefox and some versions of Internet Explorer).

(14-Apr-2008) A "/join" qualifier was added to the "search" command in the metamath program (version 0.07.37). This qualifier will join the $e hypotheses to the $a or $p for searching, so that math tokens in the $e's can be matched as well. For example, "search *com* +v" produces no results, but "search *com* +v /join" yields commutative laws involving vector addition. Thanks to Stefan Allan for suggesting this idea.

(8-Apr-2008) The 8,000th theorem, hlrel, was added to the Metamath Proof Explorer part of the database.

(2-Mar-2008) I added a small section to the end of the Deduction Theorem page.

(17-Feb-2008) ocat has uploaded the "1-Mar-2008" mmj2: mmj2.zip. See the description.

(16-Jan-2008) O'Cat has written mmj2 Proof Assistant Quick Tips.

(30-Dec-2007) "How to build a library of formalized mathematics".

(22-Dec-2007) The Metamath Proof Explorer was included in the top 30 science resources for 2007 by the University at Albany Science Library.

(17-Dec-2007) Metamath's Wikipedia entry says, "This article may require cleanup to meet Wikipedia's quality standards" (see its discussion page). Volunteers are welcome. :) (In the interest of objectivity, I don't edit this entry.)

(20-Nov-2007) Jeff Hoffman created nicod.mm and posted it to the Google Metamath Group.

(19-Nov-2007) Reinder Verlinde suggested adding tooltips to the hyperlinks on the proof pages, which I did for proof step hyperlinks. Discussion.

(5-Nov-2007) A Usenet challenge. :)

(4-Aug-2007) I added a "Request for comments on proposed 'maps to' notation" at the bottom of the AsteroidMeta set.mm discussion page.

(21-Jun-2007) A preprint (PDF file) describing Kurt Maes' axiom of choice with 5 quantifiers, proved in set.mm as ackm.

(20-Jun-2007) The 7,000th theorem, ifpr, was added to the Metamath Proof Explorer part of the database.

(29-Apr-2007) Blog mentions of Metamath: here and here.

(21-Mar-2007) Paul Chapman is working on a new proof browser, which has highlighting that allows you to see the referenced theorem before and after the substitution was made. Here is a screenshot of theorem 0nn0 and a screenshot of theorem 2p2e4.

(15-Mar-2007) A picture of Penny the cat guarding the us.metamath.org server and making the rounds.

(16-Feb-2007) For convenience, the program "drule.c" (pronounced "D-rule", not "drool") mentioned in pmproofs.txt can now be downloaded (drule.c) without having to ask me for it. The same disclaimer applies: even though this program works and has no known bugs, it was not intended for general release. Read the comments at the top of the program for instructions.

(28-Jan-2007) Jason Orendorff set up a new mailing list for Metamath: http://groups.google.com/group/metamath.

(20-Jan-2007) Bob Solovay provided a revised version of his Metamath database for Peano arithmetic, peano.mm.

(2-Jan-2007) Raph Levien has set up a wiki called Barghest for the Ghilbert language and software.

(26-Dec-2006) I posted an explanation of theorem ecoprass on Usenet.

(2-Dec-2006) Berislav Žarnić translated the Metamath Solitaire applet to Croatian.

(26-Nov-2006) Dan Getz has created an RSS feed for new theorems as they appear on this page.

(6-Nov-2006) The first 3 paragraphs in Appendix 2: Note on the Axioms were rewritten to clarify the connection between Tarski's axiom system and Metamath.

(31-Oct-2006) ocat asked for a do-over due to a bug in mmj2 -- if you downloaded the mmj2.zip version dated 10/28/2006, then download the new version dated 10/30.

(29-Oct-2006) ocat has announced that the long-awaited 1-Nov-2006 release of mmj2 is available now.
     The new "Unify+Get Hints" is quite useful, and any proof can be generated as follows. With "?" in the Hyp field and Ref field blank, select "Unify+Get Hints". Select a hint from the list and put it in the Ref field. Edit any $n dummy variables to become the desired wffs. Rinse and repeat for the new proof steps generated, until the proof is done.
     The new tutorial, mmj2PATutorial.bat, explains this in detail. One way to reduce or avoid dummy $n's is to fill in the Hyp field with a comma-separated list of any known hypothesis matches to earlier proof steps, keeping a "?" in the list to indicate that the remaining hypotheses are unknown. Then "Unify+Get Hints" can be applied. The tutorial page \mmj2\data\mmp\PATutorial\Page405.mmp has an example.
     Don't forget that the eimm export/import program lets you go back and forth between the mmj2 and the metamath program proof assistants, without exiting from either one, to exploit the best features of each as required.

(21-Oct-2006) Martin Kiselkov has written a Metamath proof verifier in the Lua scripting language, called verify.lua. While it is not practical as an everyday verifier - he writes that it takes about 40 minutes to verify set.mm on a a Pentium 4 - it could be useful to someone learning Lua or Metamath, and importantly it provides another independent way of verifying the correctness of Metamath proofs. His code looks like it is nicely structured and very readable. He is currently working on a faster version in C++.

(19-Oct-2006) New AsteroidMeta page by Raph, Distinctors_vs_binders.

(13-Oct-2006) I put a simple Metamath browser on my PDA (Palm Tungsten E) so that I don't have to lug around my laptop. Here is a screenshot. It isn't polished, but I'll provide the file + instructions if anyone wants it.

(3-Oct-2006) A blog entry, Principia for Reverse Mathematics.

(28-Sep-2006) A blog entry, Metamath responds.

(26-Sep-2006) A blog entry, Metamath isn't hygienic.

(11-Aug-2006) A blog entry, Metamath and the Peano Induction Axiom.

(26-Jul-2006) A new open problem in predicate calculus was added.

(18-Jun-2006) The 6,000th theorem, mt4d, was added to the Metamath Proof Explorer part of the database.

(9-May-2006) Luca Ciciriello has upgraded the t2mf program, which is a C program used to create the MIDI files on the Metamath Music Page, so that it works on MacOS X. This is a nice accomplishment, since the original program was written before C was standardized by ANSI and will not compile on modern compilers.
      Unfortunately, the original program source states no copyright terms. The main author, Tim Thompson, has kindly agreed to release his code to public domain, but two other authors have also contributed to the code, and so far I have been unable to contact them for copyright clearance. Therefore I cannot offer the MacOS X version for public download on this site until this is resolved. Update 10-May-2006: Another author, M. Czeiszperger, has released his contribution to public domain.
      If you are interested in Luca's modified source code, please contact me directly.

(18-Apr-2006) Incomplete proofs in progress can now be interchanged between the Metamath program's CLI Proof Assistant and mmj2's GUI Proof Assistant, using a new export-import program called eimm. This can be done without exiting either proof assistant, so that the strengths of each approach can be exploited during proof development. See "Use Case 5a" and "Use Case 5b" at mmj2ProofAssistantFeedback.

(28-Mar-2006) Scott Fenton updated his second version of Metamath Solitaire (the one that uses external axioms). He writes: "I've switched to making it a standalone program, as it seems silly to have an applet that can't be run in a web browser. Check the README file for further info." The download is mmsol-0.5.tar.gz.

(27-Mar-2006) Scott Fenton has updated the Metamath Solitaire Java applet to Java 1.5: (1) QSort has been stripped out: its functionality is in the Collections class that Sun ships; (2) all Vectors have been replaced by ArrayLists; (3) generic types have been tossed in wherever they fit: this cuts back drastically on casting; and (4) any warnings Eclipse spouted out have been dealt with. I haven't yet updated it officially, because I don't know if it will work with Microsoft's JVM in older versions of Internet Explorer. The current official version is compiled with Java 1.3, because it won't work with Microsoft's JVM if it is compiled with Java 1.4. (As distasteful as that seems, I will get complaints from users if it doesn't work with Microsoft's JVM.) If anyone can verify that Scott's new version runs on Microsoft's JVM, I would be grateful. Scott's new version is mm.java-1.5.gz; after uncompressing it, rename it to mm.java, use it to replace the existing mm.java file in the Metamath Solitaire download, and recompile according to instructions in the mm.java comments.
      Scott has also created a second version, mmsol-0.2.tar.gz, that reads the axioms from ASCII files, instead of having the axioms hard-coded in the program. This can be very useful if you want to play with custom axioms, and you can also add a collection of starting theorems as "axioms" to work from. However, it must be run from the local directory with appletviewer, since the default Java security model doesn't allow reading files from a browser. It works with the JDK 5 Update 6 Java download.
To compile (from Windows Command Prompt): C:\Program Files\Java\jdk1.5.0_06\bin\javac.exe mm.java
To run (from Windows Command Prompt): C:\Program Files\Java\jdk1.5.0_06\bin\appletviewer.exe mms.html

(21-Jan-2006) Juha Arpiainen proved the independence of axiom ax-11 from the others. This was published as an open problem in my 1995 paper (Remark 9.5 on PDF page 17). See Item 9a on the Workshop Miscellany for his seven-line proof. See also the Asteroid Meta metamathMathQuestions page under the heading "Axiom of variable substitution: ax-11". Congratulations, Juha!

(20-Oct-2005) Juha Arpiainen is working on a proof verifier in Common Lisp called Bourbaki. Its proof language has its roots in Metamath, with the goal of providing a more powerful syntax and definitional soundness checking. See its documentation and related discussion.

(17-Oct-2005) Marnix Klooster has written a Metamath proof verifier in Haskell, called Hmm. Also see his Announcement. The complete program (Hmm.hs, HmmImpl.hs, and HmmVerify.hs) has only 444 lines of code, excluding comments and blank lines. It verifies compressed as well as regular proofs; moreover, it transparently verifies both per-spec compressed proofs and the flawed format he uncovered (see comment below of 16-Oct-05).

(16-Oct-2005) Marnix Klooster noticed that for large proofs, the compressed proof format did not match the spec in the book. His algorithm to correct the problem has been put into the Metamath program (version 0.07.6). The program still verifies older proofs with the incorrect format, but the user will be nagged to update them with 'save proof *'. In set.mm, 285 out of 6376 proofs are affected. (The incorrect format did not affect proof correctness or verification, since the compression and decompression algorithms matched each other.)

(13-Sep-2005) Scott Fenton found an interesting axiom, ax46, which could be used to replace both ax-4 and ax-6.

(29-Jul-2005) Metamath was selected as site of the week by American Scientist Online.

(8-Jul-2005) Roy Longton has contributed 53 new theorems to the Quantum Logic Explorer. You can see them in the Theorem List starting at lem3.3.3lem1. He writes, "If you want, you can post an open challenge to see if anyone can find shorter proofs of the theorems I submitted."

(10-May-2005) A Usenet post I posted about the infinite prime proof; another one about indexed unions.

(3-May-2005) The theorem divexpt is the 5,000th theorem added to the Metamath Proof Explorer database.

(12-Apr-2005) Raph Levien solved the open problem in item 16 on the Workshop Miscellany page and as a corollary proved that axiom ax-9 is independent from the other axioms of predicate calculus and equality. This is the first such independence proof so far; a goal is to prove all of them independent (or to derive any redundant ones from the others).

(8-Mar-2005) I added a paragraph above our complex number axioms table, summarizing the construction and indicating where Dedekind cuts are defined. Thanks to Andrew Buhr for comments on this.

(16-Feb-2005) The Metamath Music Page is mentioned as a reference or resource for a university course called Math, Mind, and Music. .

(28-Jan-2005) Steven Cullinane parodied the Metamath Music Page in his blog.

(18-Jan-2005) Waldek Hebisch upgraded the Metamath program to run on the AMD64 64-bit processor.

(17-Jan-2005) A symbol list summary was added to the beginning of the Hilbert Space Explorer Home Page. Thanks to Mladen Pavicic for suggesting this.

(6-Jan-2005) Someone assembled an amazon.com list of some of the books in the Metamath Proof Explorer Bibliography.

(4-Jan-2005) The definition of ordinal exponentiation was decided on after this Usenet discussion.

(19-Dec-2004) A bit of trivia: my Erdös number is 2, as you can see from this list.

(20-Oct-2004) I started this Usenet discussion about the "reals are uncountable" proof (127 comments; last one on Nov. 12).

(12-Oct-2004) gch-kn shows the equivalence of the Generalized Continuum Hypothesis and Prof. Nambiar's Axiom of Combinatorial Sets. This proof answers his Open Problem 2 (PDF file).

(5-Aug-2004) I gave a talk on "Hilbert Lattice Equations" at the Argonne workshop.

(25-Jul-2004) The theorem nthruz is the 4,000th theorem added to the Metamath Proof Explorer database.

(27-May-2004) Josiah Burroughs contributed the proofs u1lemn1b, u1lem3var1, oi3oa3lem1, and oi3oa3 to the Quantum Logic Explorer database ql.mm.

(23-May-2004) Some minor typos found by Josh Purinton were corrected in the Metamath book. In addition, Josh simplified the definition of the closure of a pre-statement of a formal system in Appendix C.

(5-May-2004) Gregory Bush has found shorter proofs for 67 of the 193 propositional calculus theorems listed in Principia Mathematica, thus establishing 67 new records. (This was challenge #4 on the open problems page.)

  Copyright terms: Public domain W3C HTML validation [external]