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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.

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Last updated on 21-Nov-2024 at 5:21 AM ET.
Recent Additions to the Metamath Proof Explorer   Notes (last updated 7-Dec-2020 )
DateLabelDescription
Theorem
 
20-Nov-20242ralor 3294 Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Wolf Lammen, 20-Nov-2024.)
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓))
 
19-Nov-2024fri 5540 A nonempty subset of an 𝑅-well-founded class has an 𝑅-minimal element (inference form). (Contributed by BJ, 16-Nov-2024.) (Proof shortened by BJ, 19-Nov-2024.)
(((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
 
18-Nov-2024wfr3 8139 The principle of Well-Ordered Recursion, part 3 of 3. Finally, we show that 𝐹 is unique. We do this by showing that any function 𝐻 with the same properties we proved of 𝐹 in wfr1 8137 and wfr2 8138 is identical to 𝐹. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 18-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻)
 
18-Nov-2024wfr1 8137 The Principle of Well-Ordered Recursion, part 1 of 3. We start with an arbitrary function 𝐺. Then, using a base class 𝐴 and a set-like well-ordering 𝑅 of 𝐴, we define a function 𝐹. This function is said to be defined by "well-ordered recursion". The purpose of these three theorems is to demonstrate the properties of 𝐹. We begin by showing that 𝐹 is a function over 𝐴. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 18-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
 
18-Nov-2024wfr2a 8136 A weak version of wfr2 8138 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Scott Fenton, 30-Jul-2020.) (Proof shortened by Scott Fenton, 18-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
 
18-Nov-2024wfrresex 8135 Show without using the axiom of replacement that the restriction of the well-ordered recursion generator to a predecessor class is a set. (Contributed by Scott Fenton, 18-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)
 
18-Nov-2024csbwrecsg 8108 Move class substitution in and out of the well-founded recursive function generator. (Contributed by ML, 25-Oct-2020.) (Revised by Scott Fenton, 18-Nov-2024.)
(𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = wrecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))
 
18-Nov-2024fprresex 8097 The restriction of a function defined by well-founded recursion to the predecessor of an element of its domain is a set. Avoids the axiom of replacement. (Contributed by Scott Fenton, 18-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)
 
18-Nov-2024fprfung 8096 A "function" defined by well-founded recursion is indeed a function when the relationship is a partial order. Avoids the axiom of replacement. (Contributed by Scott Fenton, 18-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)
 
18-Nov-2024frrdmss 8094 Show without using the axiom of replacement that the domain of the well-founded recursion generator is a subclass of 𝐴. (Contributed by Scott Fenton, 18-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   dom 𝐹 ⊆ 𝐴
 
18-Nov-2024frrrel 8093 Show without using the axiom of replacement that the well-founded recursion generator gives a relation. (Contributed by Scott Fenton, 18-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   Rel 𝐹
 
18-Nov-2024fpr2 8091 Law of well-founded recursion over a partial order, part two. Now we establish the value of 𝐹 within 𝐴. (Contributed by Scott Fenton, 11-Sep-2023.) (Proof shortened by Scott Fenton, 18-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
 
18-Nov-2024fpr2a 8089 Weak version of fpr2 8091 which is useful for proofs that avoid the axiom of replacement. (Contributed by Scott Fenton, 18-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
 
18-Nov-2024csbfrecsg 8071 Move class substitution in and out of the well-founded recursive function generator. (Contributed by Scott Fenton, 18-Nov-2024.)
(𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, 𝐹) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))
 
18-Nov-2024drnf1v 2370 Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 2443 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.) Avoid ax-10 2139. (Revised by Gino Giotto, 18-Nov-2024.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))
 
18-Nov-2024dral1v 2367 Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2439 with a disjoint variable condition, which does not require ax-13 2372. Remark: the corresponding versions for dral2 2438 and drex2 2442 are instances of albidv 1924 and exbidv 1925 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2174. (Revised by Wolf Lammen, 30-Mar-2024.) Avoid ax-10 2139. (Revised by Gino Giotto, 18-Nov-2024.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
 
18-Nov-2024equsexv 2263 An equivalence related to implicit substitution. Version of equsex 2418 with a disjoint variable condition, which does not require ax-13 2372. See equsexvw 2009 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2262. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) Avoid ax-10 2139. (Revised by Gino Giotto, 18-Nov-2024.)
Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
 
17-Nov-2024bj-rdg0gALT 35169 Alternate proof of rdg0g 8229. More direct since it bypasses tz7.44-1 8208 and rdg0 8223 (and vtoclg 3495, vtoclga 3503). (Contributed by NM, 25-Apr-1995.) More direct proof. (Revised by BJ, 17-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
 
17-Nov-2024domtrfi 8938 Transitivity of dominance relation for finite sets, proved without using the Axiom of Power Sets (unlike domtr 8748). (Contributed by BTernaryTau, 17-Nov-2024.)
((𝐶 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
 
17-Nov-2024wfrfun 8134 The "function" generated by the well-ordered recursion generator is indeed a function. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 17-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)
 
17-Nov-2024wfrdmcl 8133 The predecessor class of an element of the well-ordered recursion generator's domain is a subset of its domain. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)
 
17-Nov-2024wfrdmss 8132 The domain of the well-ordered recursion generator is a subclass of 𝐴. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   dom 𝐹 ⊆ 𝐴
 
17-Nov-2024wfrrel 8131 The well-ordered recursion generator generates a relation. Avoids the axiom of replacement. (Contributed by Scott Fenton, 8-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   Rel 𝐹
 
17-Nov-2024nfwrecs 8103 Bound-variable hypothesis builder for the well-ordered recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
Ⅎ𝑥𝑅    &   Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐹    ⇒   Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹)
 
17-Nov-2024wrecseq123 8101 General equality theorem for the well-ordered recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))
 
17-Nov-2024frrdmcl 8095 Show without using the axiom of replacement that for a "function" defined by well-founded recursion, the predecessor class of an element of its domain is a subclass of its domain. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)
 
17-Nov-2024wfis2fg 6244 Well-Ordered Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
Ⅎ𝑦𝜓    &   (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    &   (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑))    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
 
17-Nov-2024wfisg 6241 Well-Ordered Induction Schema. If a property passes from all elements less than 𝑦 of a well-ordered class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 11-Feb-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
(𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
 
17-Nov-2024wfi 6238 The Principle of Well-Ordered Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if 𝐵 is a subclass of a well-ordered class 𝐴 with the property that every element of 𝐵 whose inital segment is included in 𝐴 is itself equal to 𝐴. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
(((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)
 
17-Nov-2024tz6.26 6235 All nonempty subclasses of a class having a well-ordered set-like relation have minimal elements for that relation. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
(((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
 
17-Nov-2024cbvmptv 5183 Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.) Add disjoint variable condition to avoid auxiliary axioms . See cbvmptvg 5185 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Nov-2024.)
(𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
 
17-Nov-2024cbvopab1v 5149 Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) Reduce axiom usage. (Revised by Gino Giotto, 17-Nov-2024.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
 
16-Nov-2024frd 5539 A nonempty subset of an 𝑅-well-founded class has an 𝑅-minimal element (deduction form). (Contributed by BJ, 16-Nov-2024.)
(𝜑 → 𝑅 Fr 𝐴)    &   (𝜑 → 𝐵 ⊆ 𝐴)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐵 ≠ ∅)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
 
16-Nov-2024dffr6 5538 Alternate definition of df-fr 5535. See dffr5 33627 for a definition without dummy variables (but note that their equivalence uses ax-sep 5218). (Contributed by BJ, 16-Nov-2024.)
(𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
 
14-Nov-2024aks4d1 40025 Lemma 4.1 from https://www3.nd.edu/%7eandyp/notes/AKS.pdf, existence of a polynomially bounded number by the digit size of 𝑁 that asserts the polynomial subspace that we need to search to guarantee that 𝑁 is prime. Eventually we want to show that the polynomial searching space is bounded by degree 𝐵. (Contributed by metakunt, 14-Nov-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   (𝜑 → ∃𝑟 ∈ (1...𝐵)((𝑁 gcd 𝑟) = 1 ∧ ((2 logb 𝑁)↑2) < ((odℤ‘𝑟)‘𝑁)))
 
14-Nov-2024aks4d1p9 40024 Show that the order is bound by the squared binary logarithm. (Contributed by metakunt, 14-Nov-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    ⇒   (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))
 
14-Nov-2024aks4d1lem1 39998 Technical lemma to reduce proof size. (Contributed by metakunt, 14-Nov-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵))
 
13-Nov-2024aks4d1p8d3 40022 The remainder of a division with its maximal prime power is coprime with that prime power. (Contributed by metakunt, 13-Nov-2024.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑃 ∥ 𝑁)    ⇒   (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = 1)
 
13-Nov-2024aks4d1p8d2 40021 Any prime power dividing a positive integer is less than that integer if that integer has another prime factor. (Contributed by metakunt, 13-Nov-2024.)
(𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑄 ∈ ℙ)    &   (𝜑 → 𝑃 ∥ 𝑅)    &   (𝜑 → 𝑄 ∥ 𝑅)    &   (𝜑 → ¬ 𝑃 ∥ 𝑁)    &   (𝜑 → 𝑄 ∥ 𝑁)    ⇒   (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)
 
12-Nov-2024ssdomfi 8940 A finite set dominates its subsets, proved without using the Axiom of Power Sets (unlike ssdomg 8741). (Contributed by BTernaryTau, 12-Nov-2024.)
(𝐵 ∈ Fin → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵))
 
11-Nov-2024mpteq1df 42668 An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by SN, 11-Nov-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
 
11-Nov-2024mhphf2 40209 A homogeneous polynomial defines a homogeneous function; this is mhphf 40208 with simpler notation in the conclusion in exchange for a complex definition of ∙, which is based on frlmvscafval 20883 but without the finite support restriction (frlmpws 20867, frlmbas 20872) on the assignments 𝐴 from variables to values.

TODO?: Polynomials (df-mpl 21024) are defined to have a finite amount of terms (of finite degree). As such, any assignment may be replaced by an assignment with finite support (as only a finite amount of variables matter in a given polynomial, even if the set of variables is infinite). So the finite support restriction can be assumed without loss of generality. (Contributed by SN, 11-Nov-2024.)

𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝐻 = (𝐼 mHomP 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐾 = (Base‘𝑆)    &    ∙ = ( ·𝑠 ‘((ringLMod‘𝑆) ↑s 𝐼))    &    · = (.r‘𝑆)    &    ↑ = (.g‘(mulGrp‘𝑆))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝐿 ∈ 𝑅)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
 
11-Nov-2024fldidom 20489 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.) (Proof shortened by SN, 11-Nov-2024.)
(𝑅 ∈ Field → 𝑅 ∈ IDomn)
 
11-Nov-2024ofeqd 7513 Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
(𝜑 → 𝑅 = 𝑆)    ⇒   (𝜑 → ∘f 𝑅 = ∘f 𝑆)
 
11-Nov-2024mpteq2ia 5173 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)
(𝑥 ∈ 𝐴 → 𝐵 = 𝐶)    ⇒   (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)
 
11-Nov-2024mpteq2dva 5170 Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.) Remove dependency on ax-10 2139. (Revised by SN, 11-Nov-2024.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
 
11-Nov-2024mpteq2da 5168 Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)
Ⅎ𝑥𝜑    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
 
11-Nov-2024mpteq1i 5166 An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) Remove all disjoint variable conditions. (Revised by SN, 11-Nov-2024.)
𝐴 = 𝐵    ⇒   (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
 
11-Nov-2024mpteq1 5163 An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)
(𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
 
11-Nov-2024mpteq12dva 5159 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) Remove dependency on ax-10 2139, ax-12 2173. (Revised by SN, 11-Nov-2024.)
(𝜑 → 𝐴 = 𝐶)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
 
11-Nov-2024mpteq12df 5156 An equality inference for the maps-to notation. Compare mpteq12dv 5161. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) (Proof shortened by SN, 11-Nov-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → 𝐴 = 𝐶)    &   (𝜑 → 𝐵 = 𝐷)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
 
11-Nov-2024mpteq12da 5155 An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) Remove dependency on ax-10 2139. (Revised by SN, 11-Nov-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → 𝐴 = 𝐶)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
 
10-Nov-2024aks4d1p8 40023 Show that 𝑁 and 𝑅 are coprime for AKS existence theorem, with eliminated hypothesis. (Contributed by metakunt, 10-Nov-2024.) (Proof sketch by Thierry Arnoux.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    ⇒   (𝜑 → (𝑁 gcd 𝑅) = 1)
 
10-Nov-2024aks4d1p8d1 40020 If a prime divides one number 𝑀, but not another number 𝑁, then it divides the quotient of 𝑀 and the gcd of 𝑀 and 𝑁. (Contributed by Thierry Arnoux, 10-Nov-2024.)
(𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑀)    &   (𝜑 → ¬ 𝑃 ∥ 𝑁)    ⇒   (𝜑 → 𝑃 ∥ (𝑀 / (𝑀 gcd 𝑁)))
 
9-Nov-2024bj-flddrng 35387 Fields are division rings (elemental version). (Contributed by BJ, 9-Nov-2024.)
(𝐹 ∈ Field → 𝐹 ∈ DivRing)
 
9-Nov-2024bj-dfid2ALT 35163 Alternate version of dfid2 5482. (Contributed by BJ, 9-Nov-2024.) (Proof modification is discouraged.) Use df-id 5480 instead to make the semantics of the construction df-opab 5133 clearer. (New usage is discouraged.)
I = {⟨𝑥, 𝑥⟩ ∣ ⊤}
 
7-Nov-2024ressbas 16873 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
𝑅 = (𝑊 ↾s 𝐴)    &   𝐵 = (Base‘𝑊)    ⇒   (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅))
 
7-Nov-2024setsnid 16838 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 7-Nov-2024.)
𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ 𝐷    ⇒   (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩))
 
6-Nov-2024sn-iotaex 40123 iotaex 6398 without ax-10 2139, ax-11 2156, ax-12 2173. (Contributed by SN, 6-Nov-2024.)
(℩𝑥𝜑) ∈ V
 
6-Nov-2024sn-iotassuni 40122 iotassuni 6397 without ax-10 2139, ax-11 2156, ax-12 2173. (Contributed by SN, 6-Nov-2024.)
(℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}
 
6-Nov-2024sn-iotanul 40121 Version of iotanul 6396 using df-iota 6376 instead of dfiota2 6377. (Contributed by SN, 6-Nov-2024.)
(¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
 
6-Nov-2024sn-iotauni 40120 Version of iotauni 6393 using df-iota 6376 instead of dfiota2 6377. (Contributed by SN, 6-Nov-2024.)
(∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
 
6-Nov-2024sn-iotaval 40119 Version of iotaval 6392 using df-iota 6376 instead of dfiota2 6377. (Contributed by SN, 6-Nov-2024.)
({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
 
6-Nov-2024sn-iotalemcor 40118 Corollary of sn-iotalem 40117. Compare sb8iota 6388. (Contributed by SN, 6-Nov-2024.)
(℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦})
 
6-Nov-2024sn-iotalem 40117 An unused lemma showing that many equivalences involving df-iota 6376 are potentially provable without ax-10 2139, ax-11 2156, ax-12 2173. (Contributed by SN, 6-Nov-2024.)
{𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
 
6-Nov-2024eqimssd 40111 Equality implies inclusion, deduction version. (Contributed by SN, 6-Nov-2024.)
(𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → 𝐴 ⊆ 𝐵)
 
6-Nov-2024hlhilsmul 39885 Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &    · = (.r‘𝐸)    ⇒   (𝜑 → · = (.r‘𝑅))
 
6-Nov-2024hlhilsplus 39883 Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &    + = (+g‘𝐸)    ⇒   (𝜑 → + = (+g‘𝑅))
 
6-Nov-2024hlhilsbase 39881 The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   𝐶 = (Base‘𝐸)    ⇒   (𝜑 → 𝐶 = (Base‘𝑅))
 
6-Nov-2024hlhilslem 39879 Lemma for hlhilsbase 39881 etc. (Contributed by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   𝐹 = Slot (𝐹‘ndx)    &   (𝐹‘ndx) ≠ (*𝑟‘ndx)    &   𝐶 = (𝐹‘𝐸)    ⇒   (𝜑 → 𝐶 = (𝐹‘𝑅))
 
6-Nov-2024oppradd 19786 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (oppr‘𝑅)    &    + = (+g‘𝑅)    ⇒    + = (+g‘𝑂)
 
6-Nov-2024opprbas 19784 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (oppr‘𝑅)    &   𝐵 = (Base‘𝑅)    ⇒   𝐵 = (Base‘𝑂)
 
6-Nov-2024opprlem 19782 Lemma for opprbas 19784 and oppradd 19786. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
𝑂 = (oppr‘𝑅)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (.r‘ndx)    ⇒   (𝐸‘𝑅) = (𝐸‘𝑂)
 
6-Nov-2024symgvalstruct 18919 The value of the symmetric group function at 𝐴 represented as extensible structure with three slots. This corresponds to the former definition of SymGrp. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 31-Mar-2024.) (Proof shortened by AV, 6-Nov-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}    &   𝑀 = (𝐴 ↑m 𝐴)    &    + = (𝑓 ∈ 𝑀, 𝑔 ∈ 𝑀 ↦ (𝑓 ∘ 𝑔))    &   𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))    ⇒   (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
 
6-Nov-2024frmdplusg 18408 The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) (Proof shortened by AV, 6-Nov-2024.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝑀)    &    + = (+g‘𝑀)    ⇒    + = ( ++ ↾ (𝐵 × 𝐵))
 
5-Nov-2024dfid2 5482 Alternate definition of the identity relation. Instance of dfid3 5483 not requiring auxiliary axioms. (Contributed by NM, 15-Mar-2007.) Reduce axiom usage. (Revised by Gino Giotto, 4-Nov-2024.) (Proof shortened by BJ, 5-Nov-2024.)

Use df-id 5480 instead to make the semantics of the constructor df-opab 5133 clearer. (New usage is discouraged.)

I = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥}
 
5-Nov-2024r19.30 3265 Restricted quantifier version of 19.30 1885. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 5-Nov-2024.)
(∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
 
4-Nov-2024sbthfi 8942 Schroeder-Bernstein Theorem for finite sets, proved without using the Axiom of Power Sets (unlike sbth 8833). (Contributed by BTernaryTau, 4-Nov-2024.)
((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)
 
4-Nov-2024sbthfilem 8941 Lemma for sbthfi 8942. (Contributed by BTernaryTau, 4-Nov-2024.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    &   𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))    &   𝐵 ∈ V    ⇒   ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)
 
4-Nov-2024r19.29vva 3263 A commonly used pattern based on r19.29 3183, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)    ⇒   (𝜑 → 𝜒)
 
4-Nov-2024r19.29d2r 3261 Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)    &   (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))
 
4-Nov-2024r19.12 3252 Restricted quantifier version of 19.12 2325. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2372, ax-ext 2709. (Revised by Wolf Lammen, 17-Jun-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
 
4-Nov-2024ralrexbid 3250 Formula-building rule for restricted existential quantifier, using a restricted universal quantifier to bind the quantified variable in the antecedent. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
(𝜑 → (𝜓 ↔ 𝜃))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃))
 
4-Nov-2024reximdvai 3199 Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 8-Jan-2020.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
(𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))
 
4-Nov-2024exexw 2055 Existential quantification over a given variable is idempotent. Weak version of bj-exexbiex 34809, requiring fewer axioms. (Contributed by Gino Giotto, 4-Nov-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥𝜑 ↔ ∃𝑥∃𝑥𝜑)
 
3-Nov-2024znmul 20660 The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℕ0 → (.r‘𝑈) = (.r‘𝑌))
 
3-Nov-2024znadd 20658 The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℕ0 → (+g‘𝑈) = (+g‘𝑌))
 
3-Nov-2024znbas2 20656 The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑌))
 
3-Nov-2024znbaslem 20654 Lemma for znbas 20663. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 9-Sep-2021.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (le‘ndx)    ⇒   (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌))
 
3-Nov-2024zlmmulr 20636 Ring operation of a ℤ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &    · = (.r‘𝐺)    ⇒    · = (.r‘𝑊)
 
3-Nov-2024zlmplusg 20634 Group operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &    + = (+g‘𝐺)    ⇒    + = (+g‘𝑊)
 
3-Nov-2024zlmbas 20632 Base set of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐵 = (Base‘𝐺)    ⇒   𝐵 = (Base‘𝑊)
 
3-Nov-2024zlmlem 20630 Lemma for zlmbas 20632 and zlmplusg 20634. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (Scalar‘ndx)    &   (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)    ⇒   (𝐸‘𝐺) = (𝐸‘𝑊)
 
3-Nov-2024nelb 3194 A definition of ¬ 𝐴 ∈ 𝐵. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.) (Proof shortened by Wolf Lammen, 3-Nov-2024.)
(¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
 
3-Nov-2024rexbi 3169 Distribute restricted quantification over a biconditional. (Contributed by Scott Fenton, 7-Aug-2024.) (Proof shortened by Wolf Lammen, 3-Nov-2024.)
(∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓))
 
2-Nov-2024psrvscafval 21069 The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 2-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &    ∙ = ( ·𝑠 ‘𝑆)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r‘𝑅)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    ⇒    ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓))
 
2-Nov-2024zlmsca 20638 Scalar ring of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → ℤring = (Scalar‘𝑊))
 
2-Nov-2024rexab 3624 Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) Reduce axiom usage. (Revised by Gino Giotto, 2-Nov-2024.)
(𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒))
 
2-Nov-2024ralab 3621 Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) Reduce axiom usage. (Revised by Gino Giotto, 2-Nov-2024.)
(𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒))
 
1-Nov-2024mnringvscad 41731 The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐵 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐵)    &   (𝜑 → 𝑅 ∈ 𝑈)    &   (𝜑 → 𝑀 ∈ 𝑊)    ⇒   (𝜑 → ( ·𝑠 ‘𝑉) = ( ·𝑠 ‘𝐹))
 
1-Nov-2024mnringscad 41729 The scalar ring of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   (𝜑 → 𝑅 ∈ 𝑈)    &   (𝜑 → 𝑀 ∈ 𝑊)    ⇒   (𝜑 → 𝑅 = (Scalar‘𝐹))
 
1-Nov-2024mnringaddgd 41724 The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐴 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐴)    &   (𝜑 → 𝑅 ∈ 𝑈)    &   (𝜑 → 𝑀 ∈ 𝑊)    ⇒   (𝜑 → (+g‘𝑉) = (+g‘𝐹))
 
1-Nov-2024mnringbased 41718 The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐴 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐴)    &   𝐵 = (Base‘𝑉)    &   (𝜑 → 𝑅 ∈ 𝑈)    &   (𝜑 → 𝑀 ∈ 𝑊)    ⇒   (𝜑 → 𝐵 = (Base‘𝐹))
 
1-Nov-2024mnringnmulrd 41716 Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (Revised by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (.r‘ndx)    &   𝐴 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐴)    &   (𝜑 → 𝑅 ∈ 𝑈)    &   (𝜑 → 𝑀 ∈ 𝑊)    ⇒   (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))
 
1-Nov-2024opsrsca 21170 The scalar ring of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    ⇒   (𝜑 → 𝑅 = (Scalar‘𝑂))
 
1-Nov-2024opsrvsca 21168 The scalar product operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   (𝜑 → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑂))
 
1-Nov-2024opsrmulr 21166 The multiplication operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   (𝜑 → (.r‘𝑆) = (.r‘𝑂))
 
1-Nov-2024opsrplusg 21164 The addition operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   (𝜑 → (+g‘𝑆) = (+g‘𝑂))
 
1-Nov-2024opsrbas 21162 The base set of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   (𝜑 → (Base‘𝑆) = (Base‘𝑂))
 
1-Nov-2024opsrbaslem 21160 Get a component of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 9-Sep-2021.) (Revised by AV, 1-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (le‘ndx)    ⇒   (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂))
 
1-Nov-2024plendxnvscandx 17008 The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. Formerly part of proof for opsrvsca 21168. (Contributed by AV, 1-Nov-2024.)
(le‘ndx) ≠ ( ·𝑠 ‘ndx)
 
1-Nov-2024plendxnscandx 17007 The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. Formerly part of proof for opsrsca 21170. (Contributed by AV, 1-Nov-2024.)
(le‘ndx) ≠ (Scalar‘ndx)
 
1-Nov-2024plendxnmulrndx 17006 The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. Formerly part of proof for opsrmulr 21166. (Contributed by AV, 1-Nov-2024.)
(le‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024mendvscafval 40931 Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
𝐴 = (MEndo‘𝑀)    &    · = ( ·𝑠 ‘𝑀)    &   𝐵 = (Base‘𝐴)    &   𝑆 = (Scalar‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝐸 = (Base‘𝑀)    ⇒   ( ·𝑠 ‘𝐴) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))
 
31-Oct-2024mendsca 40930 The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
𝐴 = (MEndo‘𝑀)    &   𝑆 = (Scalar‘𝑀)    ⇒   𝑆 = (Scalar‘𝐴)
 
31-Oct-2024mendmulrfval 40928 Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)    ⇒   (.r‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))
 
31-Oct-2024mendplusgfval 40926 Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)    &    + = (+g‘𝑀)    ⇒   (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))
 
31-Oct-2024aks4d1p7 40019 Technical step in AKS lemma 4.1 (Contributed by metakunt, 31-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    ⇒   (𝜑 → ∃𝑝 ∈ ℙ (𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁))
 
31-Oct-2024aks4d1p7d1 40018 Technical step in AKS lemma 4.1 (Contributed by metakunt, 31-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    &   (𝜑 → ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁))    ⇒   (𝜑 → 𝑅 ∥ (𝑁↑(⌊‘(2 logb 𝐵))))
 
31-Oct-2024ttrclse 33713 If 𝑅 is set-like over 𝐴, then the transitive closure of the restriction of 𝑅 to 𝐴 is set-like over 𝐴.

This theorem requires the axioms of infinity and replacement for its proof. (Contributed by Scott Fenton, 31-Oct-2024.)

(𝑅 Se 𝐴 → t++(𝑅 ↾ 𝐴) Se 𝐴)
 
31-Oct-2024ttrclselem2 33712 Lemma for ttrclse 33713. Show that a suc 𝑁 element long chain gives membership in the 𝑁-th predecessor class and vice-versa. (Contributed by Scott Fenton, 31-Oct-2024.)
𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))    ⇒   ((𝑁 ∈ ω ∧ 𝑅 Se 𝐴 ∧ 𝑋 ∈ 𝐴) → (∃𝑓(𝑓 Fn suc suc 𝑁 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘suc 𝑁) = 𝑋) ∧ ∀𝑎 ∈ suc 𝑁(𝑓‘𝑎)(𝑅 ↾ 𝐴)(𝑓‘suc 𝑎)) ↔ 𝑦 ∈ (𝐹‘𝑁)))
 
31-Oct-2024ttrclselem1 33711 Lemma for ttrclse 33713. Show that all finite ordinal function values of 𝐹 are subsets of 𝐴. (Contributed by Scott Fenton, 31-Oct-2024.)
𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))    ⇒   (𝑁 ∈ ω → (𝐹‘𝑁) ⊆ 𝐴)
 
31-Oct-2024rdg0n 33598 If 𝐴 is a proper class, then the recursive function generator at ∅ is the empty set. (Contributed by Scott Fenton, 31-Oct-2024.)
(¬ 𝐴 ∈ V → (rec(𝐹, 𝐴)‘∅) = ∅)
 
31-Oct-2024resvmulr 31440 .r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝐻 = (𝐺 ↾v 𝐴)    &    · = (.r‘𝐺)    ⇒   (𝐴 ∈ 𝑉 → · = (.r‘𝐻))
 
31-Oct-2024resvvsca 31438 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Proof shortened by AV, 31-Oct-2024.)
𝐻 = (𝐺 ↾v 𝐴)    &    · = ( ·𝑠 ‘𝐺)    ⇒   (𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘𝐻))
 
31-Oct-2024resvplusg 31436 +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝐻 = (𝐺 ↾v 𝐴)    &    + = (+g‘𝐺)    ⇒   (𝐴 ∈ 𝑉 → + = (+g‘𝐻))
 
31-Oct-2024resvbas 31434 Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝐻 = (𝐺 ↾v 𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ 𝑉 → 𝐵 = (Base‘𝐻))
 
31-Oct-2024resvlem 31432 Other elements of a scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝑅 = (𝑊 ↾v 𝐴)    &   𝐶 = (𝐸‘𝑊)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (Scalar‘ndx)    ⇒   (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅))
 
31-Oct-2024nrgtrg 23760 A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.) (Proof shortened by AV, 31-Oct-2024.)
(𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)
 
31-Oct-2024tngip 23715 The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    , = (·𝑖‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → , = (·𝑖‘𝑇))
 
31-Oct-2024tngvsca 23713 The scalar multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    · = ( ·𝑠 ‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → · = ( ·𝑠 ‘𝑇))
 
31-Oct-2024tngsca 23711 The scalar ring of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐹 = (Scalar‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → 𝐹 = (Scalar‘𝑇))
 
31-Oct-2024tngmulr 23709 The ring multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    · = (.r‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → · = (.r‘𝑇))
 
31-Oct-2024tng0 23708 The group identity of a structure augmented with a norm. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    0 = (0g‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → 0 = (0g‘𝑇))
 
31-Oct-2024tngplusg 23706 The group addition of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    + = (+g‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → + = (+g‘𝑇))
 
31-Oct-2024tngbas 23704 The base set of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐵 = (Base‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → 𝐵 = (Base‘𝑇))
 
31-Oct-2024tnglem 23702 Lemma for tngbas 23704 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (TopSet‘ndx)    &   (𝐸‘ndx) ≠ (dist‘ndx)    ⇒   (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇))
 
31-Oct-2024indistpsALT 22071 The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 22069 from the direct component assignment version indistps2 22070. (Contributed by NM, 24-Oct-2012.) (Revised by AV, 31-Oct-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩}    ⇒   𝐾 ∈ TopSp
 
31-Oct-2024eltpsg 22000 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by AV, 31-Oct-2024.)
𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)
 
31-Oct-2024dsndxnmulrndx 17022 The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(dist‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024tsetndxnmulrndx 16993 The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(TopSet‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024tsetndxnbasendx 16991 The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.)
(TopSet‘ndx) ≠ (Base‘ndx)
 
31-Oct-2024basendxlttsetndx 16990 The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(Base‘ndx) < (TopSet‘ndx)
 
31-Oct-2024tsetndxnn 16989 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.)
(TopSet‘ndx) ∈ ℕ
 
31-Oct-2024oveqprc 16821 Lemma for showing the equality of values for functions like slot extractors 𝐸 at a proper class. Extracted from several former proofs of lemmas like resvlem 31432. (Contributed by AV, 31-Oct-2024.)
(𝐸‘∅) = ∅    &   𝑍 = (𝑋𝑂𝑌)    &   Rel dom 𝑂    ⇒   (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑍))
 
31-Oct-2024fveqprc 16820 Lemma for showing the equality of values for functions like slot extractors 𝐸 at a proper class. Extracted from several former proofs of lemmas like zlmlem 20630. (Contributed by AV, 31-Oct-2024.)
(𝐸‘∅) = ∅    &   𝑌 = (𝐹‘𝑋)    ⇒   (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑌))
 
31-Oct-2024reximia 3172 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Wolf Lammen, 31-Oct-2024.)
(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
 
31-Oct-2024ralcom4 3161 Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom dependencies. (Revised by BJ, 13-Jun-2019.) (Proof shortened by Wolf Lammen, 31-Oct-2024.)
(∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)
 
31-Oct-2024ralbida 3156 Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.) (Proof shortened by Wolf Lammen, 31-Oct-2024.)
Ⅎ𝑥𝜑    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
 
31-Oct-2024nfra2w 3151 Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 42369. Version of nfra2 3154 with a disjoint variable condition not requiring ax-13 2372. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by Gino Giotto, 24-Sep-2024.) (Proof shortened by Wolf Lammen, 31-Oct-2024.)
Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
 
30-Oct-2024aks4d1p6 40017 The maximal prime power exponent is smaller than the binary logarithm floor of 𝐵. (Contributed by metakunt, 30-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑃 ∥ 𝑅)    &   𝐾 = (𝑃 pCnt 𝑅)    ⇒   (𝜑 → 𝐾 ≤ (⌊‘(2 logb 𝐵)))
 
30-Oct-2024aks4d1p5 40016 Show that 𝑁 and 𝑅 are coprime for AKS existence theorem. Precondition will be eliminated in further theorem. (Contributed by metakunt, 30-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    &   (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)    ⇒   (𝜑 → (𝑁 gcd 𝑅) = 1)
 
30-Oct-2024baseltedgf 27266 The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 30-Oct-2024.)
(Base‘ndx) < (.ef‘ndx)
 
30-Oct-2024isposix 17958 Properties that determine a poset (explicit structure version). Note that the numeric indices of the structure components are not mentioned explicitly in either the theorem or its proof. (Contributed by NM, 9-Nov-2012.) (Proof shortened by AV, 30-Oct-2024.)
𝐵 ∈ V    &    ≤ ∈ V    &   𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ≤ ⟩}    &   (𝑥 ∈ 𝐵 → 𝑥 ≤ 𝑥)    &   ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))    &   ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))    ⇒   𝐾 ∈ Poset
 
30-Oct-2024plendxnbasendx 17004 The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.)
(le‘ndx) ≠ (Base‘ndx)
 
30-Oct-2024basendxltplendx 17003 The index value of the Base slot is less than the index value of the le slot. (Contributed by AV, 30-Oct-2024.)
(Base‘ndx) < (le‘ndx)
 
30-Oct-2024plendxnn 17002 The index value of the order slot is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 30-Oct-2024.)
(le‘ndx) ∈ ℕ
 
30-Oct-2024pm13.181 3025 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Oct-2024.)
((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
 
29-Oct-2024cchhllem 27157 Lemma for chlbas and chlvsca . (Contributed by Thierry Arnoux, 15-Apr-2019.) (Revised by AV, 29-Oct-2024.)
𝐶 = (((subringAlg ‘ℂfld)‘ℝ) sSet ⟨(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))⟩)    &   𝐸 = Slot (𝐸‘ndx)    &   (Scalar‘ndx) ≠ (𝐸‘ndx)    &   ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)    &   (·𝑖‘ndx) ≠ (𝐸‘ndx)    ⇒   (𝐸‘ℂfld) = (𝐸‘𝐶)
 
29-Oct-2024ttgds 27150 The metric of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.)
𝐺 = (toTG‘𝐻)    &   𝐷 = (dist‘𝐻)    ⇒   𝐷 = (dist‘𝐺)
 
29-Oct-2024ttgvsca 27148 The scalar product of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.)
𝐺 = (toTG‘𝐻)    &    · = ( ·𝑠 ‘𝐻)    ⇒    · = ( ·𝑠 ‘𝐺)
 
29-Oct-2024ttgplusg 27145 The addition operation of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.)
𝐺 = (toTG‘𝐻)    &    + = (+g‘𝐻)    ⇒    + = (+g‘𝐺)
 
29-Oct-2024ttgbas 27143 The base set of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.)
𝐺 = (toTG‘𝐻)    &   𝐵 = (Base‘𝐻)    ⇒   𝐵 = (Base‘𝐺)
 
29-Oct-2024ttglem 27141 Lemma for ttgbas 27143, ttgvsca 27148 etc. (Contributed by Thierry Arnoux, 15-Apr-2019.) (Revised by AV, 29-Oct-2024.)
𝐺 = (toTG‘𝐻)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (LineG‘ndx)    &   (𝐸‘ndx) ≠ (Itv‘ndx)    ⇒   (𝐸‘𝐻) = (𝐸‘𝐺)
 
29-Oct-2024slotslnbpsd 26708 The slots Base, +g, ·𝑠 and dist are different from the slot LineG. Formerly part of ttglem 27141 and proofs using it. (Contributed by AV, 29-Oct-2024.)
(((LineG‘ndx) ≠ (Base‘ndx) ∧ (LineG‘ndx) ≠ (+g‘ndx)) ∧ ((LineG‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (LineG‘ndx) ≠ (dist‘ndx)))
 
29-Oct-2024slotsinbpsd 26707 The slots Base, +g, ·𝑠 and dist are different from the slot Itv. Formerly part of ttglem 27141 and proofs using it. (Contributed by AV, 29-Oct-2024.)
(((Itv‘ndx) ≠ (Base‘ndx) ∧ (Itv‘ndx) ≠ (+g‘ndx)) ∧ ((Itv‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (Itv‘ndx) ≠ (dist‘ndx)))
 
29-Oct-2024tngds 23717 The metric function of a structure augmented with a norm. (Contributed by Mario Carneiro, 3-Oct-2015.) (Proof shortened by AV, 29-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    − = (-g‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) = (dist‘𝑇))
 
29-Oct-2024srads 20368 Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (dist‘𝑊) = (dist‘𝐴))
 
29-Oct-2024sratset 20365 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴))
 
29-Oct-2024sramulr 20360 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (.r‘𝑊) = (.r‘𝐴))
 
29-Oct-2024sraaddg 20358 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (+g‘𝑊) = (+g‘𝐴))
 
29-Oct-2024srabase 20356 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (Base‘𝑊) = (Base‘𝐴))
 
29-Oct-2024sralem 20354 Lemma for srabase 20356 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   𝐸 = Slot (𝐸‘ndx)    &   (Scalar‘ndx) ≠ (𝐸‘ndx)    &   ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)    &   (·𝑖‘ndx) ≠ (𝐸‘ndx)    ⇒   (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))
 
29-Oct-2024dsndxntsetndx 17024 The slot for the distance function is not the slot for the topology in an extensible structure. Formerly part of proof for tngds 23717. (Contributed by AV, 29-Oct-2024.)
(dist‘ndx) ≠ (TopSet‘ndx)
 
29-Oct-2024slotsdnscsi 17023 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot dist. Formerly part of sralem 20354 and proofs using it. (Contributed by AV, 29-Oct-2024.)
((dist‘ndx) ≠ (Scalar‘ndx) ∧ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (dist‘ndx) ≠ (·𝑖‘ndx))
 
29-Oct-2024slotstnscsi 16994 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. Formerly part of sralem 20354 and proofs using it. (Contributed by AV, 29-Oct-2024.)
((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx))
 
29-Oct-2024ipndxnmulrndx 16970 The slot for the inner product is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for mgpsca 19643. (Contributed by AV, 29-Oct-2024.)
(·𝑖‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024ipndxnplusgndx 16969 The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(·𝑖‘ndx) ≠ (+g‘ndx)
 
29-Oct-2024vscandxnmulrndx 16959 The slot for the scalar product is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for rmodislmod 20106. (Contributed by AV, 29-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024scandxnmulrndx 16954 The slot for the scalar field is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for mgpsca 19643. (Contributed by AV, 29-Oct-2024.)
(Scalar‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024pm13.18 3024 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 29-Oct-2024.)
((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶)
 
28-Oct-2024aks4d1p4 40015 There exists a small enough number such that it does not divide 𝐴. (Contributed by metakunt, 28-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    ⇒   (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
 
28-Oct-2024edgfndxid 27264 The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 28-Oct-2024.)
(𝐺 ∈ 𝑉 → (.ef‘𝐺) = (𝐺‘(.ef‘ndx)))
 
28-Oct-2024tuslem 23326 Lemma for tusbas 23328, tusunif 23329, and tustopn 23331. (Contributed by Thierry Arnoux, 5-Dec-2017.) (Proof shortened by AV, 28-Oct-2024.)
𝐾 = (toUnifSp‘𝑈)    ⇒   (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))
 
28-Oct-2024estrreslem1 17769 Lemma 1 for estrres 17772. (Contributed by AV, 14-Mar-2020.) (Proof shortened by AV, 28-Oct-2024.)
(𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    ⇒   (𝜑 → 𝐵 = (Base‘𝐶))
 
28-Oct-2024slotsbhcdif 17044 The slots Base, Hom and comp are different. (Contributed by AV, 5-Mar-2020.) (Proof shortened by AV, 28-Oct-2024.)
((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx))
 
28-Oct-2024unifndxntsetndx 17030 The slot for the uniform set is not the slot for the topology in an extensible structure. Formerly part of proof for tuslem 23326. (Contributed by AV, 28-Oct-2024.)
(UnifSet‘ndx) ≠ (TopSet‘ndx)
 
28-Oct-2024basendxltunifndx 17028 The index of the slot for the base set is less then the index of the slot for the uniform set in an extensible structure. Formerly part of proof for tuslem 23326. (Contributed by AV, 28-Oct-2024.)
(Base‘ndx) < (UnifSet‘ndx)
 
28-Oct-2024unifndxnn 17027 The index of the slot for the uniform set in an extensible structure is a positive integer. Formerly part of proof for tuslem 23326. (Contributed by AV, 28-Oct-2024.)
(UnifSet‘ndx) ∈ ℕ
 
28-Oct-2024dsndxnbasendx 17020 The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.)
(dist‘ndx) ≠ (Base‘ndx)
 
28-Oct-2024basendxltdsndx 17019 The index of the slot for the base set is less then the index of the slot for the distance in an extensible structure. Formerly part of proof for tmslem 23543. (Contributed by AV, 28-Oct-2024.)
(Base‘ndx) < (dist‘ndx)
 
28-Oct-2024dsndxnn 17018 The index of the slot for the distance in an extensible structure is a positive integer. Formerly part of proof for tmslem 23543. (Contributed by AV, 28-Oct-2024.)
(dist‘ndx) ∈ ℕ
 
28-Oct-2024basendxnmulrndx 16931 The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) (Proof shortened by AV, 28-Oct-2024.)
(Base‘ndx) ≠ (.r‘ndx)
 
28-Oct-2024wunress 16886 Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 28-Oct-2024.)
(𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑 → 𝑊 ∈ 𝑈)    ⇒   (𝜑 → (𝑊 ↾s 𝐴) ∈ 𝑈)
 
28-Oct-2024predpo 6215 Property of the predecessor class for partial orders. (Contributed by Scott Fenton, 28-Apr-2012.) (Proof shortened by Scott Fenton, 28-Oct-2024.)
((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋)))
 
28-Oct-2024predtrss 6214 If 𝑅 is transitive over 𝐴 and 𝑌𝑅𝑋, then Pred(𝑅, 𝐴, 𝑌) is a subclass of Pred(𝑅, 𝐴, 𝑋). (Contributed by Scott Fenton, 28-Oct-2024.)
((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))
 
28-Oct-2024necon3ai 2967 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 28-Oct-2024.)
(𝜑 → 𝐴 = 𝐵)    ⇒   (𝐴 ≠ 𝐵 → ¬ 𝜑)
 
28-Oct-2024sbabel 2940 Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 28-Oct-2024.)
Ⅎ𝑥𝐴    ⇒   ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)
 
27-Oct-2024aks4d1p3 40014 There exists a small enough number such that it does not divide 𝐴. (Contributed by metakunt, 27-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)
 
27-Oct-2024aks4d1p2 40013 Technical lemma for existence of non-divisor. (Contributed by metakunt, 27-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   (𝜑 → (2↑𝐵) ≤ (lcm‘(1...𝐵)))
 
27-Oct-2024grpplusg 16924 The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by AV, 27-Oct-2024.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ( + ∈ 𝑉 → + = (+g‘𝐺))
 
27-Oct-2024grpbase 16922 The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by AV, 27-Oct-2024.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
 
27-Oct-2024grpstrndx 16921 A constructed group is a structure. Version not depending on the implementation of the indices. (Contributed by AV, 27-Oct-2024.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   𝐺 Struct ⟨(Base‘ndx), (+g‘ndx)⟩
 
27-Oct-2024df-wrecs 8099 Define the well-ordered recursive function generator. This function takes the usual expressions from recursion theorems and forms a unified definition. Specifically, given a function 𝐹, a relation 𝑅, and a base set 𝐴, this definition generates a function 𝐺 = wrecs(𝑅, 𝐴, 𝐹) that has property that, at any point 𝑥 ∈ 𝐴, (𝐺‘𝑥) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑥))). See wfr1 8137, wfr2 8138, and wfr3 8139. (Contributed by Scott Fenton, 7-Jun-2018.) (Revised by BJ, 27-Oct-2024.)
wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
 
27-Oct-2024opco2 7936 Value of an operation precomposed with the projection on the second component. (Contributed by BJ, 27-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (𝐴(𝐹 ∘ 2nd )𝐵) = (𝐹‘𝐵))
 
27-Oct-2024opco1 7935 Value of an operation precomposed with the projection on the first component. (Contributed by Mario Carneiro, 28-May-2014.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (𝐴(𝐹 ∘ 1st )𝐵) = (𝐹‘𝐴))
 
27-Oct-2024predexg 6209 The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.)
(𝐴 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V)
 
26-Oct-2024sticksstones22 40052 Non-exhaustive sticks and stones. (Contributed by metakunt, 26-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑆 ∈ Fin)    &   (𝜑 → 𝑆 ≠ ∅)    &   𝐴 = {𝑓 ∣ (𝑓:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑓‘𝑖) ≤ 𝑁)}    ⇒   (𝜑 → (♯‘𝐴) = ((𝑁 + (♯‘𝑆))C(♯‘𝑆)))
 
26-Oct-2024dfttrcl2 33710 When 𝑅 is a set and a relationship, then its transitive closure can be defined by an intersection. (Contributed by Scott Fenton, 26-Oct-2024.)
((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → t++𝑅 = ∩ {𝑧 ∣ (𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
 
26-Oct-2024ttrclexg 33709 If 𝑅 is a set, then so is t++𝑅. (Contributed by Scott Fenton, 26-Oct-2024.)
(𝑅 ∈ 𝑉 → t++𝑅 ∈ V)
 
26-Oct-2024rnttrcl 33708 The range of a transitive closure is the same as the range of the original class. (Contributed by Scott Fenton, 26-Oct-2024.)
ran t++𝑅 = ran 𝑅
 
26-Oct-2024dmttrcl 33707 The domain of a transitive closure is the same as the domain of the original class. (Contributed by Scott Fenton, 26-Oct-2024.)
dom t++𝑅 = dom 𝑅
 
26-Oct-2024nfttrcld 33696 Bound variable hypothesis builder for transitive closure. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.)
(𝜑 → Ⅎ𝑥𝑅)    ⇒   (𝜑 → Ⅎ𝑥t++𝑅)
 
26-Oct-2024nfopab 5139 Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.) (Revised by Scott Fenton, 26-Oct-2024.)
Ⅎ𝑧𝜑    ⇒   Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
26-Oct-2024nfopabd 5138 Bound-variable hypothesis builder for class abstraction. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑧𝜓)    ⇒   (𝜑 → Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜓})
 
26-Oct-2024sbceqal 3778 Class version of one implication of equvelv 2035. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by SN, 26-Oct-2024.)
(𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))
 
26-Oct-2024sbcim1 3767 Distribution of class substitution over implication. One direction of sbcimg 3762 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) Avoid ax-10 2139, ax-12 2173. (Revised by SN, 26-Oct-2024.)
([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
 
26-Oct-2024sbievg 2361 Substitution applied to expressions linked by implicit substitution. The proof was part of a former cbvabw 2813 version. (Contributed by GG and WL, 26-Oct-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
 
25-Oct-2024hbab1 2724 Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2024.)
(𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
 
25-Oct-2024nfsbv 2328 If 𝑧 is not free in 𝜑, then it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is disjoint from both 𝑥 and 𝑦. Version of nfsb 2527 with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2372. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) (Proof shortened by Wolf Lammen, 25-Oct-2024.)
Ⅎ𝑧𝜑    ⇒   Ⅎ𝑧[𝑦 / 𝑥]𝜑
 
24-Oct-2024sticksstones21 40051 Lift sticks and stones to arbitrary finite non-empty sets. (Contributed by metakunt, 24-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑆 ∈ Fin)    &   (𝜑 → 𝑆 ≠ ∅)    &   𝐴 = {𝑓 ∣ (𝑓:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑓‘𝑖) = 𝑁)}    ⇒   (𝜑 → (♯‘𝐴) = ((𝑁 + ((♯‘𝑆) − 1))C((♯‘𝑆) − 1)))
 
24-Oct-2024sticksstones20 40050 Lift sticks and stones to arbitrary finite non-empty sets. (Contributed by metakung, 24-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑆 ∈ Fin)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}    &   (𝜑 → (♯‘𝑆) = 𝐾)    ⇒   (𝜑 → (♯‘𝐵) = ((𝑁 + (𝐾 − 1))C(𝐾 − 1)))
 
24-Oct-2024eldifsucnn 33597 Condition for membership in the difference of ω and a nonzero finite ordinal. (Contributed by Scott Fenton, 24-Oct-2024.)
(𝐴 ∈ ω → (𝐵 ∈ (ω ∖ suc 𝐴) ↔ ∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥))
 
24-Oct-2024eqtr3 2764 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Wolf Lammen, 24-Oct-2024.)
((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
 
24-Oct-2024eqtr2 2762 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 24-Oct-2024.)
((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)
 
23-Oct-2024sticksstones19 40049 Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}    &   (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))    ⇒   (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
 
23-Oct-2024sticksstones18 40048 Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}    &   (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))    ⇒   (𝜑 → 𝐹:𝐴⟶𝐵)
 
23-Oct-2024sticksstones17 40047 Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}    &   (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)    &   𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))    ⇒   (𝜑 → 𝐺:𝐵⟶𝐴)
 
23-Oct-2024eqeq12 2755 Equality relationship among four classes. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2024.)
((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
 
23-Oct-2024eqeq12d 2754 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Oct-2024.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    ⇒   (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
 
23-Oct-2024eqeqan12d 2752 A useful inference for substituting definitions into an equality. See also eqeqan12dALT 2760. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) Shorten other proofs. (Revised by Wolf Lammen, 23-Oct-2024.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜓 → 𝐶 = 𝐷)    ⇒   ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
 
21-Oct-2024unifndxnbasendx 17029 The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(UnifSet‘ndx) ≠ (Base‘ndx)
 
21-Oct-2024ipndxnbasendx 16968 The slot for the inner product is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(·𝑖‘ndx) ≠ (Base‘ndx)
 
21-Oct-2024scandxnbasendx 16952 The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(Scalar‘ndx) ≠ (Base‘ndx)
 
20-Oct-2024sticksstones16 40046 Sticks and stones with collapsed definitions for positive integers. (Contributed by metakunt, 20-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    ⇒   (𝜑 → (♯‘𝐴) = ((𝑁 + (𝐾 − 1))C(𝐾 − 1)))
 
20-Oct-2024ttrclss 33706 If 𝑅 is a subclass of 𝑆 and 𝑆 is transitive, then the transitive closure of 𝑅 is a subclass of 𝑆. (Contributed by Scott Fenton, 20-Oct-2024.)
((𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → t++𝑅 ⊆ 𝑆)
 
20-Oct-2024cottrcl 33705 Composition law for the transitive closure of a relationship. (Contributed by Scott Fenton, 20-Oct-2024.)
(𝑅 ∘ t++𝑅) ⊆ t++𝑅
 
20-Oct-2024ttrclco 33704 Composition law for the transitive closure of a relationship. (Contributed by Scott Fenton, 20-Oct-2024.)
(t++𝑅 ∘ 𝑅) ⊆ t++𝑅
 
20-Oct-2024ttrclresv 33703 The transitive closure of 𝑅 restricted to V is the same as the transitive closure of 𝑅 itself. (Contributed by Scott Fenton, 20-Oct-2024.)
t++(𝑅 ↾ V) = t++𝑅
 
19-Oct-2024resseqnbas 16877 The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
𝑅 = (𝑊 ↾s 𝐴)    &   𝐶 = (𝐸‘𝑊)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (Base‘ndx)    ⇒   (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅))
 
18-Oct-2024rmodislmod 20106 The right module 𝑅 induces a left module 𝐿 by replacing the scalar multiplication with a reversed multiplication if the scalar ring is commutative. The hypothesis "rmodislmod.r" is a definition of a right module analogous to Definition df-lmod 20040 of a left module, see also islmod 20042. (Contributed by AV, 3-Dec-2021.) (Proof shortened by AV, 18-Oct-2024.)
𝑉 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = ( ·𝑠 ‘𝑅)    &   𝐹 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐹)    &    ⨣ = (+g‘𝐹)    &    × = (.r‘𝐹)    &    1 = (1r‘𝐹)    &   (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑤 · 𝑟) ∈ 𝑉 ∧ ((𝑤 + 𝑥) · 𝑟) = ((𝑤 · 𝑟) + (𝑥 · 𝑟)) ∧ (𝑤 · (𝑞 ⨣ 𝑟)) = ((𝑤 · 𝑞) + (𝑤 · 𝑟))) ∧ ((𝑤 · (𝑞 × 𝑟)) = ((𝑤 · 𝑞) · 𝑟) ∧ (𝑤 · 1 ) = 𝑤)))    &    ∗ = (𝑠 ∈ 𝐾, 𝑣 ∈ 𝑉 ↦ (𝑣 · 𝑠))    &   𝐿 = (𝑅 sSet ⟨( ·𝑠 ‘ndx), ∗ ⟩)    ⇒   (𝐹 ∈ CRing → 𝐿 ∈ LMod)
 
18-Oct-2024mgpress 19650 Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)
𝑆 = (𝑅 ↾s 𝐴)    &   𝑀 = (mulGrp‘𝑅)    ⇒   ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (mulGrp‘𝑆))
 
18-Oct-2024setsplusg 18869 The other components of an extensible structure remain unchanged if the +g component is set/substituted. (Contributed by Stefan O'Rear, 26-Aug-2015.) Generalisation of the former oppglem and mgplem. (Revised by AV, 18-Oct-2024.)
𝑂 = (𝑅 sSet ⟨(+g‘ndx), 𝑆⟩)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (+g‘ndx)    ⇒   (𝐸‘𝑅) = (𝐸‘𝑂)
 
18-Oct-2024rescbas 17458 Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 18-Oct-2024.)
𝐷 = (𝐶 ↾cat 𝐻)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   (𝜑 → 𝑆 = (Base‘𝐷))
 
18-Oct-2024oppcbas 17345 Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 18-Oct-2024.)
𝑂 = (oppCat‘𝐶)    &   𝐵 = (Base‘𝐶)    ⇒   𝐵 = (Base‘𝑂)
 
18-Oct-2024dsndxnplusgndx 17021 The slot for the distance function is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpds 19648. (Contributed by AV, 18-Oct-2024.)
(dist‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024plendxnplusgndx 17005 The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. Formerly part of proof for oppgle 31140. (Contributed by AV, 18-Oct-2024.)
(le‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024tsetndxnplusgndx 16992 The slot for the topology is not the slot for the group operation in an extensible structure. Formerly part of proof for oppgtset 18873. (Contributed by AV, 18-Oct-2024.)
(TopSet‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024vscandxnscandx 16960 The slot for the scalar product is not the slot for the scalar field in an extensible structure. Formerly part of proof for rmodislmod 20106. (Contributed by AV, 18-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (Scalar‘ndx)
 
18-Oct-2024vscandxnplusgndx 16958 The slot for the scalar product is not the slot for the group operation in an extensible structure. Formerly part of proof for rmodislmod 20106. (Contributed by AV, 18-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024vscandxnbasendx 16957 The slot for the scalar product is not the slot for the base set in an extensible structure. Formerly part of proof for rmodislmod 20106. (Contributed by AV, 18-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (Base‘ndx)
 
18-Oct-2024scandxnplusgndx 16953 The slot for the scalar field is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpsca 19643. (Contributed by AV, 18-Oct-2024.)
(Scalar‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024starvndxnmulrndx 16942 The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 16944. (Contributed by AV, 18-Oct-2024.)
(*𝑟‘ndx) ≠ (.r‘ndx)
 
18-Oct-2024starvndxnplusgndx 16941 The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 16944. (Contributed by AV, 18-Oct-2024.)
(*𝑟‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024starvndxnbasendx 16940 The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 16944. (Contributed by AV, 18-Oct-2024.)
(*𝑟‘ndx) ≠ (Base‘ndx)
 
17-Oct-2024ttrcltr 33702 The transitive closure of a class is transitive. (Contributed by Scott Fenton, 17-Oct-2024.)
(t++𝑅 ∘ t++𝑅) ⊆ t++𝑅
 
17-Oct-2024ssttrcl 33701 If 𝑅 is a relation, then it is a subclass of its transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
(Rel 𝑅 → 𝑅 ⊆ t++𝑅)
 
17-Oct-2024relttrcl 33698 The transitive closure of a class is a relation. (Contributed by Scott Fenton, 17-Oct-2024.)
Rel t++𝑅
 
17-Oct-2024nfttrcl 33697 Bound variable hypothesis builder for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
Ⅎ𝑥𝑅    ⇒   Ⅎ𝑥t++𝑅
 
17-Oct-2024ttrcleq 33695 Equality theorem for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
(𝑅 = 𝑆 → t++𝑅 = t++𝑆)
 
17-Oct-2024df-ttrcl 33694 Define the transitive closure of a class. This is the smallest relationship containing 𝑅 (or more precisely, the relation (𝑅 ↾ V) induced by 𝑅) and having the transitive property. Definition from [Levy] p. 59, who denotes it as 𝑅∗ and calls it the "ancestral" of 𝑅. (Contributed by Scott Fenton, 17-Oct-2024.)
t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚))}
 
17-Oct-2024nnasmo 33596 Finite ordinal subtraction cancels on the left. (Contributed by Scott Fenton, 17-Oct-2024.)
(𝐴 ∈ ω → ∃*𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
 
17-Oct-2024nnuni 33595 The union of a finite ordinal is a finite ordinal. (Contributed by Scott Fenton, 17-Oct-2024.)
(𝐴 ∈ ω → ∪ 𝐴 ∈ ω)
 
17-Oct-2024basendxnplusgndx 16918 The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 17-Oct-2024.)
(Base‘ndx) ≠ (+g‘ndx)
 
17-Oct-2024basendxltplusgndx 16917 The index of the slot for the base set is less then the index of the slot for the group operation in an extensible structure. (Contributed by AV, 17-Oct-2024.)
(Base‘ndx) < (+g‘ndx)
 
17-Oct-2024plusgndxnn 16916 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.)
(+g‘ndx) ∈ ℕ
 
17-Oct-2024ressval3d 16882 Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.) (Proof shortened by AV, 17-Oct-2024.)
𝑅 = (𝑆 ↾s 𝐴)    &   𝐵 = (Base‘𝑆)    &   𝐸 = (Base‘ndx)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → 𝐸 ∈ dom 𝑆)    &   (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
 
17-Oct-20242strstr1 16863 A constructed two-slot structure. Version of 2strstr 16860 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Proof shortened by AV, 17-Oct-2024.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   (Base‘ndx) < 𝑁    &   𝑁 ∈ ℕ    ⇒   𝐺 Struct ⟨(Base‘ndx), 𝑁⟩
 
17-Oct-20241strwun 16858 A constructed one-slot structure in a weak universe. (Contributed by AV, 27-Mar-2020.) (Proof shortened by AV, 17-Oct-2024.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    ⇒   ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)
 
17-Oct-2024basndxelwund 16852 The index of the base set is an element in a weak universe containing the natural numbers. Formerly part of proof for 1strwun 16858. (Contributed by AV, 27-Mar-2020.) (Revised by AV, 17-Oct-2024.)
(𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    ⇒   (𝜑 → (Base‘ndx) ∈ 𝑈)
 
17-Oct-2024setsidvald 16828 Value of the structure replacement function, deduction version.

Hint: Do not substitute 𝑁 by a specific (positive) integer to be independent of a hard-coded index value. Often, (𝐸‘ndx) can be used instead of 𝑁. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 17-Oct-2024.)

𝐸 = Slot 𝑁    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → 𝑁 ∈ dom 𝑆)    ⇒   (𝜑 → 𝑆 = (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩))
 
16-Oct-2024thincciso 46218 Two thin categories are isomorphic iff the induced preorders are order-isomorphic. Example 3.26(2) of [Adamek] p. 33. (Contributed by Zhi Wang, 16-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑅 = (Base‘𝑋)    &   𝑆 = (Base‘𝑌)    &   𝐻 = (Hom ‘𝑋)    &   𝐽 = (Hom ‘𝑌)    &   (𝜑 → 𝑈 ∈ 𝑉)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑋 ∈ ThinCat)    &   (𝜑 → 𝑌 ∈ ThinCat)    ⇒   (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)))
 
16-Oct-2024bj-elabd2ALT 35040 Alternate proof of elabd2 3594 bypassing elab6g 3593 (and using sbiedvw 2098 instead of the ∀𝑥(𝑥 = 𝑦 → 𝜓) idiom). (Contributed by BJ, 16-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 = {𝑥 ∣ 𝜓})    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒))
 
16-Oct-2024omsinds 7708 Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.) (Proof shortened by BJ, 16-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   (𝐴 ∈ ω → 𝜒)
 
16-Oct-2024predon 7612 The predecessor of an ordinal under E and On is itself. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by BJ, 16-Oct-2024.)
(𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
 
16-Oct-2024elpred 6208 Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.) (Proof shortened by BJ, 16-Oct-2024.)
𝑌 ∈ V    ⇒   (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
 
16-Oct-2024elpredim 6207 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.) (Proof shortened by BJ, 16-Oct-2024.)
𝑋 ∈ V    ⇒   (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)
 
16-Oct-2024elpredimg 6206 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.) Generalize to closed form. (Revised by BJ, 16-Oct-2024.)
((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
 
16-Oct-2024elpredg 6205 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) (Proof shortened by BJ, 16-Oct-2024.)
((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
 
16-Oct-2024elpredgg 6204 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) Generalize to closed form. (Revised by BJ, 16-Oct-2024.)
((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
 
16-Oct-2024epin 5992 Any set is equal to its preimage under the converse membership relation. (Contributed by Mario Carneiro, 9-Mar-2013.) Put in closed form. (Revised by BJ, 16-Oct-2024.)
(𝐴 ∈ 𝑉 → (◡ E “ {𝐴}) = 𝐴)
 
16-Oct-2024elinisegg 5990 Membership in the inverse image of a singleton. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Put in closed form and shorten. (Revised by BJ, 16-Oct-2024.)
((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
 
16-Oct-2024elimasn 5986 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by BJ, 16-Oct-2024.) TODO: replace existing usages by usages of elimasn1 5984, remove, and relabel elimasn1 5984 to "elimasn".
𝐵 ∈ V    &   𝐶 ∈ V    ⇒   (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
 
16-Oct-2024elimasn1 5984 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Use df-br 5071 and shorten. (Revised by BJ, 16-Oct-2024.)
𝐵 ∈ V    &   𝐶 ∈ V    ⇒   (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
 
16-Oct-2024elimasng1 5983 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) Revise to use df-br 5071 and to prove elimasn1 5984 from it. (Revised by BJ, 16-Oct-2024.)
((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
 
16-Oct-2024elabd2 3594 Membership in a class abstraction, using implicit substitution. Deduction version of elab 3602. (Contributed by Gino Giotto, 12-Oct-2024.) (Revised by BJ, 16-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 = {𝑥 ∣ 𝜓})    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒))
 
15-Oct-2024eloprabga 7360 The law of concretion for operation class abstraction. Compare elopab 5433. (Contributed by NM, 14-Sep-1999.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.) Avoid ax-10 2139, ax-11 2156. (Revised by Wolf Lammen, 15-Oct-2024.)
((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
 
15-Oct-2024cbvopabv 5143 Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.) Reduce axiom usage. (Revised by Gino Giotto, 15-Oct-2024.)
((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
 
15-Oct-2024notabw 4234 A class abstraction defined by a negation. Version of notab 4235 using implicit substitution, which does not require ax-10 2139, ax-12 2173. (Contributed by Gino Giotto, 15-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜓})
 
15-Oct-2024unabw 4228 Union of two class abstractions. Version of unab 4229 using implicit substitution, which does not require ax-8 2110, ax-10 2139, ax-12 2173. (Contributed by Gino Giotto, 15-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    ⇒   ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑦 ∣ (𝜒 ∨ 𝜃)}
 
15-Oct-2024csbied 3866 Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.) Reduce axiom usage. (Revised by Gino Giotto, 15-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    ⇒   (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
 
15-Oct-2024csbie 3864 Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.) Reduce axiom usage. (Revised by Gino Giotto, 15-Oct-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
 
15-Oct-2024csbconstg 3847 Substitution doesn't affect a constant 𝐵 (in which 𝑥 does not occur). csbconstgf 3846 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.) Avoid ax-12 2173. (Revised by Gino Giotto, 15-Oct-2024.)
(𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
 
14-Oct-2024catcxpccl 17840 The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑇 = (𝑋 ×c 𝑌)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → 𝑇 ∈ 𝐵)
 
14-Oct-2024catcfuccl 17750 The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑄 = (𝑋 FuncCat 𝑌)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → 𝑄 ∈ 𝐵)
 
14-Oct-2024catcccocl 17747 The composition operation of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 17748. (Contributed by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (comp‘𝑋) ∈ 𝑈)
 
14-Oct-2024catchomcl 17746 The Hom-set of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 17748. (Contributed by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
 
14-Oct-2024catcbaselcl 17745 The base set of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 17748. (Contributed by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (Base‘𝑋) ∈ 𝑈)
 
14-Oct-2024catcslotelcl 17744 A slot entry of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 17748. (Contributed by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   𝐸 = Slot (𝐸‘ndx)    ⇒   (𝜑 → (𝐸‘𝑋) ∈ 𝑈)
 
14-Oct-2024catcbascl 17743 An element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 17748. (Contributed by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝑋 ∈ 𝑈)
 
14-Oct-2024fuchom 17594 The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    ⇒   𝑁 = (Hom ‘𝑄)
 
14-Oct-2024oppchomfval 17340 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
𝐻 = (Hom ‘𝐶)    &   𝑂 = (oppCat‘𝐶)    ⇒   tpos 𝐻 = (Hom ‘𝑂)
 
13-Oct-2024edgfndxnn 27263 The index value of the edge function extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 13-Oct-2024.)
(.ef‘ndx) ∈ ℕ
 
13-Oct-2024edgfndx 27262 Index value of the df-edgf 27260 slot. (Contributed by AV, 13-Oct-2024.) (New usage is discouraged.)
(.ef‘ndx) = 18
 
13-Oct-20240pos 17954 Technical lemma to simplify the statement of ipopos 18169. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 16818) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Proof shortened by AV, 13-Oct-2024.)
∅ ∈ Poset
 
13-Oct-2024catcoppccl 17748 The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑂 = (oppCat‘𝑋)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝑂 ∈ 𝐵)
 
13-Oct-2024wunnat 17588 A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
(𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝐶 ∈ 𝑈)    &   (𝜑 → 𝐷 ∈ 𝑈)    ⇒   (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)
 
13-Oct-2024wunfunc 17530 A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
(𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝐶 ∈ 𝑈)    &   (𝜑 → 𝐷 ∈ 𝑈)    ⇒   (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
 
13-Oct-2024rescco 17462 Composition in the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
𝐷 = (𝐶 ↾cat 𝐻)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   (𝜑 → 𝑆 ⊆ 𝐵)    &    · = (comp‘𝐶)    ⇒   (𝜑 → · = (comp‘𝐷))
 
13-Oct-2024prdsvallem 17082 Lemma for prdsval 17083. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 17083, dependency on df-hom 16912 removed. (Revised by AV, 13-Oct-2024.)
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V
 
13-Oct-2024basendxnn 16850 The index value of the base set extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 13-Oct-2024.)
(Base‘ndx) ∈ ℕ
 
13-Oct-2024pcxnn0cl 16489 Extended nonnegative integer closure of the general prime count function. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 pCnt 𝑁) ∈ ℕ0*)
 
12-Oct-2024trpred 6223 The class of predecessors of an element of a transitive class for the membership relation is that element. (Contributed by BJ, 12-Oct-2024.)
((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)
 
12-Oct-2024sbc2ie 3795 Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Gino Giotto, 12-Oct-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)
 
12-Oct-2024sbcg 3791 Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3789. (Contributed by Alan Sare, 10-Nov-2012.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)
(𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
 
12-Oct-2024sbcimdv 3786 Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1814). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)
(𝜑 → (𝜓 → 𝜒))    ⇒   (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))
 
12-Oct-2024sbcied 3756 Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) Avoid ax-10 2139, ax-12 2173. (Revised by Gino Giotto, 12-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
 
12-Oct-2024sbcieg 3751 Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2139, ax-12 2173. (Revised by Gino Giotto, 12-Oct-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
 
12-Oct-2024elabgt 3596 Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3600.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)
((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
 
12-Oct-2024elabd3 3595 Membership in a class abstraction, using implicit substitution. Deduction version of elab 3602. (Contributed by Gino Giotto, 12-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒))
 
12-Oct-2024ceqsexv 3469 Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) Avoid ax-12 2173. (Revised by Gino Giotto, 12-Oct-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
 
12-Oct-2024rabrabi 3417 Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2139, ax-11 2156 and ax-12 2173. (Revised by Gino Giotto, 12-Oct-2024.)
(𝑥 = 𝑦 → (𝜒 ↔ 𝜑))    ⇒   {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)}
 
12-Oct-2024sbco4lem 2276 Lemma for sbco4 2278. It replaces the temporary variable 𝑣 with another temporary variable 𝑤. (Contributed by Jim Kingdon, 26-Sep-2018.) (Proof shortened by Wolf Lammen, 12-Oct-2024.)
([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
 
11-Oct-2024sticksstones12a 40041 Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 11-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → ∀𝑑 ∈ 𝐵 (𝐹‘(𝐺‘𝑑)) = 𝑑)
 
10-Oct-2024dfuniv2 41809 Alternative definition of Univ using only simple defined symbols. (Contributed by Rohan Ridenour, 10-Oct-2024.)
Univ = {𝑦 ∣ ∀𝑧 ∈ 𝑦 ∀𝑓 ∈ 𝒫 𝑦∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ (𝑦 ∩ 𝑤) ∧ (𝑧 ∩ ∪ 𝑓) ⊆ ∪ (𝑓 ∩ 𝒫 𝒫 𝑤))}
 
10-Oct-2024ismnushort 41808 Express the predicate on 𝑈 and 𝑧 in ismnu 41768 in a shorter form while avoiding complicated definitions. (Contributed by Rohan Ridenour, 10-Oct-2024.)
(∀𝑓 ∈ 𝒫 𝑈∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ (𝑈 ∩ 𝑤) ∧ (𝑧 ∩ ∪ 𝑓) ⊆ ∪ (𝑓 ∩ 𝒫 𝒫 𝑤)) ↔ (𝒫 𝑧 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
 
9-Oct-2024no3inds 34042 Triple induction over surreal numbers. (Contributed by Scott Fenton, 9-Oct-2024.)
(𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (((∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜃 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜒 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜁) ∧ (∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))𝜓 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜏 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜎) ∧ ∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜂) → 𝜑))    ⇒   ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No ) → 𝜆)
 
9-Oct-2024lrold 34004 The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 9-Oct-2024.)
(( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))
 
9-Oct-2024rightirr 34001 No surreal is a member of its right set. (Contributed by Scott Fenton, 9-Oct-2024.)
¬ 𝑋 ∈ ( R ‘𝑋)
 
9-Oct-2024leftirr 34000 No surreal is a member of its left set. (Contributed by Scott Fenton, 9-Oct-2024.)
¬ 𝑋 ∈ ( L ‘𝑋)
 
9-Oct-2024madeun 33993 The made set is the union of the old set and the new set. (Contributed by Scott Fenton, 9-Oct-2024.)
( M ‘𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴))
 
9-Oct-2024rightssno 33991 The right set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( R ‘𝐴) ⊆ No
 
9-Oct-2024leftssno 33990 The left set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( L ‘𝐴) ⊆ No
 
9-Oct-2024rightssold 33989 The right options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
( R ‘𝑋) ⊆ ( O ‘( bday ‘𝑋))
 
9-Oct-2024leftssold 33988 The left options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
( L ‘𝑋) ⊆ ( O ‘( bday ‘𝑋))
 
9-Oct-2024oldssmade 33987 The older-than set is a subset of the made set. (Contributed by Scott Fenton, 9-Oct-2024.)
( O ‘𝐴) ⊆ ( M ‘𝐴)
 
9-Oct-2024madess 33986 If 𝐴 is less than or equal to ordinal 𝐵, then the made set of 𝐴 is included in the made set of 𝐵. (Contributed by Scott Fenton, 9-Oct-2024.)
((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐴) ⊆ ( M ‘𝐵))
 
9-Oct-2024rightval 33975 The value of the right options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}
 
9-Oct-2024leftval 33974 The value of the left options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}
 
9-Oct-2024newssno 33973 New sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( N ‘𝐴) ⊆ No
 
9-Oct-2024oldssno 33972 Old sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( O ‘𝐴) ⊆ No
 
9-Oct-2024madessno 33971 Made sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( M ‘𝐴) ⊆ No
 
9-Oct-2024newval 33966 The value of the new options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))
 
9-Oct-2024dtru 5288 At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Note that we may not substitute the same variable for both 𝑥 and 𝑦 (as indicated by the distinct variable requirement), for otherwise we would contradict stdpc6 2032.

This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2709 or ax-sep 5218. See dtruALT 5306 for a shorter proof using these axioms.

The proof makes use of dummy variables 𝑧 and 𝑤 which do not appear in the final theorem. They must be distinct from each other and from 𝑥 and 𝑦. In other words, if we were to substitute 𝑥 for 𝑧 throughout the proof, the proof would fail. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2372. (Revised by Gino Giotto, 5-Sep-2023.) Avoid ax-12 2173. (Revised by Rohan Ridenour, 9-Oct-2024.)

¬ ∀𝑥 𝑥 = 𝑦
 
8-Oct-2024rr-grothshort 41811 A shorter equivalent of ax-groth 10510 than rr-groth 41806 using a few more simple defined symbols. (Contributed by Rohan Ridenour, 8-Oct-2024.)
∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∀𝑓 ∈ 𝒫 𝑦∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ (𝑦 ∩ 𝑤) ∧ (𝑧 ∩ ∪ 𝑓) ⊆ ∪ (𝑓 ∩ 𝒫 𝒫 𝑤)))
 
8-Oct-2024rr-grothshortbi 41810 Express "every set is contained in a Grothendieck universe" in a short form while avoiding complicated definitions. (Contributed by Rohan Ridenour, 8-Oct-2024.)
(∀𝑥∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ∀𝑥∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∀𝑓 ∈ 𝒫 𝑦∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ (𝑦 ∩ 𝑤) ∧ (𝑧 ∩ ∪ 𝑓) ⊆ ∪ (𝑓 ∩ 𝒫 𝒫 𝑤))))
 
7-Oct-2024f1ocof1ob2 44461 If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is bijective iff 𝐹 and 𝐺 are both bijective. Symmetric version of f1ocof1ob 44460 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 7-Oct-2024.)
((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ (𝐹:𝐴–1-1-onto→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷)))
 
7-Oct-2024f1ocof1ob 44460 If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is bijective iff 𝐹 and 𝐺 are both bijective. (Contributed by GL and AV, 7-Oct-2024.)
((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷)))
 
7-Oct-2024fcoresf1ob 44454 A composition is bijective iff the restriction of its first component to the minimum domain is bijective and the restriction of its second component to the minimum domain is injective. (Contributed by GL and AV, 7-Oct-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    ⇒   (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1-onto→𝐷)))
 
7-Oct-2024fcoresfob 44453 A composition is surjective iff the restriction of its first component to the minimum domain is surjective. (Contributed by GL and AV, 7-Oct-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    ⇒   (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))
 
7-Oct-2024fcoresf1b 44451 A composition is injective iff the restrictions of its components to the minimum domains are injective. (Contributed by GL and AV, 7-Oct-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    ⇒   (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))
 
7-Oct-2024fcoresf1 44450 If a composition is injective, then the restrictions of its components to the minimum domains are injective. (Contributed by GL and AV, 18-Sep-2024.) (Revised by AV, 7-Oct-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    &   (𝜑 → (𝐺 ∘ 𝐹):𝑃–1-1→𝐷)    ⇒   (𝜑 → (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷))
 
7-Oct-2024sticksstones15 40045 Sticks and stones with almost collapsed definitions for positive integers. (Contributed by metakunt, 7-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    ⇒   (𝜑 → (♯‘𝐴) = ((𝑁 + 𝐾)C𝐾))
 
7-Oct-2024sticksstones14 40044 Sticks and stones with definitions as hypotheses. (Contributed by metakunt, 7-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → (♯‘𝐴) = ((𝑁 + 𝐾)C𝐾))
 
7-Oct-2024bj-sbievwd 34891 Variant of sbievw 2097. (Contributed by BJ, 7-Oct-2024.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → Ⅎ'𝑥𝜒)    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
 
7-Oct-2024bj-equsexvwd 34890 Variant of equsexvw 2009. (Contributed by BJ, 7-Oct-2024.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → Ⅎ'𝑥𝜒)    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒))
 
7-Oct-2024bj-equsalvwd 34889 Variant of equsalvw 2008. (Contributed by BJ, 7-Oct-2024.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → Ⅎ'𝑥𝜒)    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))
 
7-Oct-2024bj-equsvt 34888 A variant of equsv 2007. (Contributed by BJ, 7-Oct-2024.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
 
7-Oct-2024bj-pm11.53a 34887 A variant of pm11.53v 1948. One can similarly prove a variant with DV (𝑦, 𝜑) and ∀𝑦Ⅎ'𝑥𝜓 instead of DV (𝑥, 𝜓) and ∀𝑥Ⅎ'𝑦𝜑. (Contributed by BJ, 7-Oct-2024.)
(∀𝑥Ⅎ'𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
 
7-Oct-2024bj-pm11.53v 34886 Version of pm11.53v 1948 with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024.)
((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
 
7-Oct-2024bj-pm11.53vw 34885 Version of pm11.53v 1948 with nonfreeness antecedents. One can also prove the theorem with antecedent (Ⅎ'𝑦∀𝑥𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓). (Contributed by BJ, 7-Oct-2024.)
((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
 
6-Oct-2024sticksstones13 40043 Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
 
6-Oct-2024sticksstones12 40042 Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
 
6-Oct-2024sticksstones11 40040 Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 = 0)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
 
6-Oct-2024sticksstones10 40039 Establish mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐺:𝐵⟶𝐴)
 
6-Oct-2024sticksstones9 40038 Establish mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 = 0)    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐺:𝐵⟶𝐴)
 
6-Oct-2024nfnbi 1858 A variable is nonfree in a proposition if and only if it is so in its negation. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 6-Oct-2024.)
(Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)
 
5-Oct-2024bj-ntrufal 34677 The negation of a theorem is equivalent to false. This can shorten dfnul2 4256. (Contributed by BJ, 5-Oct-2024.)
𝜑    ⇒   (¬ 𝜑 ↔ ⊥)
 
5-Oct-2024bj-mt2bi 34676 Version of mt2 199 where the major premise is a biconditional. Another proof is also possible via con2bii 357 and mpbi 229. The current mt2bi 363 should be relabeled, maybe to imfal. (Contributed by BJ, 5-Oct-2024.)
𝜑    &   (𝜓 ↔ ¬ 𝜑)    ⇒    ¬ 𝜓
 
5-Oct-2024sbc6g 3741 An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by SN, 5-Oct-2024.)
(𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
 
5-Oct-2024elab 3602 Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) Avoid ax-10 2139, ax-11 2156, ax-12 2173. (Revised by SN, 5-Oct-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
 
5-Oct-2024elabg 3600 Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2372. (Revised by SN, 23-Nov-2022.) Avoid ax-10 2139, ax-11 2156, ax-12 2173. (Revised by SN, 5-Oct-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
 
5-Oct-2024elab6g 3593 Membership in a class abstraction. Class version of sb6 2089. (Contributed by SN, 5-Oct-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
 
4-Oct-2024bj-gabima 35055 Generalized class abstraction as a direct image.

TODO: improve the support lemmas elimag 5962 and fvelima 6817 to nonfreeness hypothesis (and for the latter, biconditional). (Contributed by BJ, 4-Oct-2024.)

(𝜑 → ∀𝑥𝜑)    &   (𝜑 → Ⅎ𝑥𝐹)    &   (𝜑 → Fun 𝐹)    &   (𝜑 → {𝑥 ∣ 𝜓} ⊆ dom 𝐹)    ⇒   (𝜑 → {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} = (𝐹 “ {𝑥 ∣ 𝜓}))
 
4-Oct-2024bj-elgab 35054 Elements of a generalized class abstraction. (Contributed by BJ, 4-Oct-2024.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → (∃𝑥(𝐴 = 𝐵 ∧ 𝜓) ↔ 𝜒))    ⇒   (𝜑 → (𝐴 ∈ {𝐵 ∣ 𝑥 ∣ 𝜓} ↔ 𝜒))
 
4-Oct-2024bj-gabeqis 35053 Equality of generalized class abstractions, with implicit substitution. (Contributed by BJ, 4-Oct-2024.)
(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝐵 ∣ 𝑦 ∣ 𝜓}
 
4-Oct-2024bj-gabeqd 35052 Equality of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → 𝐴 = 𝐵)    &   (𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} = {𝐵 ∣ 𝑥 ∣ 𝜒})
 
4-Oct-2024bj-gabssd 35051 Inclusion of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → 𝐴 = 𝐵)    &   (𝜑 → (𝜓 → 𝜒))    ⇒   (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒})
 
4-Oct-2024bj-gabss 35050 Inclusion of generalized class abstractions. (Contributed by BJ, 4-Oct-2024.)
(∀𝑥(𝐴 = 𝐵 ∧ (𝜑 → 𝜓)) → {𝐴 ∣ 𝑥 ∣ 𝜑} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜓})
 
4-Oct-2024df-bj-gab 35049 Definition of generalized class abstractions: typically, 𝑥 is a bound variable in 𝐴 and 𝜑 and {𝐴 ∣ 𝑥 ∣ 𝜑} denotes "the class of 𝐴(𝑥)'s such that 𝜑(𝑥)". (Contributed by BJ, 4-Oct-2024.)
{𝐴 ∣ 𝑥 ∣ 𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)}
 
3-Oct-2024itg1addlem4 24768 Lemma for itg1add 24771. (Contributed by Mario Carneiro, 28-Jun-2014.) (Proof shortened by SN, 3-Oct-2024.)
(𝜑 → 𝐹 ∈ dom ∫1)    &   (𝜑 → 𝐺 ∈ dom ∫1)    &   𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))))    &   𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))    ⇒   (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
 
3-Oct-2024peano5 7714 The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43, except our proof does not require the Axiom of Infinity. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as Theorem findes 7723. (Contributed by NM, 18-Feb-2004.) Avoid ax-10 2139, ax-12 2173. (Revised by Gino Giotto, 3-Oct-2024.)
((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
 
3-Oct-2024dffr2 5544 Alternate definition of well-founded relation. Similar to Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Mario Carneiro, 23-Jun-2015.) Avoid ax-10 2139, ax-11 2156, ax-12 2173, but use ax-8 2110. (Revised by Gino Giotto, 3-Oct-2024.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
 
3-Oct-2024pocl 5501 Characteristic properties of a partial order in class notation. (Contributed by NM, 27-Mar-1997.) Reduce axiom usage and shorten proof. (Revised by Gino Giotto, 3-Oct-2024.)
(𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
 
3-Oct-20240nelopab 5471 The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) Reduce axiom usage and shorten proof. (Revised by Gino Giotto, 3-Oct-2024.)
¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
3-Oct-2024ab0w 4304 The class of sets verifying a property is the empty class if and only if that property is a contradiction. Version of ab0 4305 using implicit substitution, which requires fewer axioms. (Contributed by Gino Giotto, 3-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝜓)
 
3-Oct-2024vtocl3ga 3507 Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.) Reduce axiom usage. (Revised by Gino Giotto, 3-Oct-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    &   ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑)    ⇒   ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃)
 
3-Oct-2024vtocl3g 3501 Implicit substitution of a class for a setvar variable. Version of vtocl3gf 3499 with disjoint variable conditions instead of nonfreeness hypotheses, requiring fewer axioms. (Contributed by Gino Giotto, 3-Oct-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    &   𝜑    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃)
 
3-Oct-2024rspw 3128 Restricted specialization. Weak version of rsp 3129, requiring ax-8 2110, but not ax-12 2173. (Contributed by Gino Giotto, 3-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
 
1-Oct-2024thincfth 46217 A functor from a thin category is faithful. (Contributed by Zhi Wang, 1-Oct-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
 
1-Oct-2024fullthinc2 46216 A full functor to a thin category maps empty hom-sets to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.)
𝐵 = (Base‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐷 ∈ ThinCat)    &   (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋𝐻𝑌) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
 
1-Oct-2024fullthinc 46215 A functor to a thin category is full iff empty hom-sets are mapped to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.)
𝐵 = (Base‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐷 ∈ ThinCat)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)))
 
1-Oct-2024functhinc 46214 A functor to a thin category is determined entirely by the object part. The hypothesis "functhinc.1" is related to a monotone function if preorders induced by the categories are considered (catprs2 46181). (Contributed by Zhi Wang, 1-Oct-2024.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   (𝜑 → 𝐷 ∈ Cat)    &   (𝜑 → 𝐸 ∈ ThinCat)    &   (𝜑 → 𝐹:𝐵⟶𝐶)    &   𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    &   (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))    ⇒   (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ 𝐺 = 𝐾))
 
1-Oct-2024functhinclem4 46213 Lemma for functhinc 46214. Other requirements on the morphism part are automatically satisfied. (Contributed by Zhi Wang, 1-Oct-2024.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   (𝜑 → 𝐷 ∈ Cat)    &   (𝜑 → 𝐸 ∈ ThinCat)    &   (𝜑 → 𝐹:𝐵⟶𝐶)    &   𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    &   (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))    &    1 = (Id‘𝐷)    &   𝐼 = (Id‘𝐸)    &    · = (comp‘𝐷)    &   𝑂 = (comp‘𝐸)    ⇒   ((𝜑 ∧ 𝐺 = 𝐾) → ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘( 1 ‘𝑎)) = (𝐼‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑚 ∈ (𝑎𝐻𝑏)∀𝑛 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑛(⟨𝑎, 𝑏⟩ · 𝑐)𝑚)) = (((𝑏𝐺𝑐)‘𝑛)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩𝑂(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑚))))
 
1-Oct-2024functhinclem3 46212 Lemma for functhinc 46214. The mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.)
(𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))    &   (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))))    &   (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))    &   (𝜑 → ∃*𝑛 𝑛 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))    ⇒   (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
 
1-Oct-2024functhinclem2 46211 Lemma for functhinc 46214. (Contributed by Zhi Wang, 1-Oct-2024.)
(𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))    ⇒   (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))
 
1-Oct-2024functhinclem1 46210 Lemma for functhinc 46214. Given the object part, there is only one possible morphism part such that the mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   (𝜑 → 𝐸 ∈ ThinCat)    &   (𝜑 → 𝐹:𝐵⟶𝐶)    &   𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    &   ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))    ⇒   (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤))) ↔ 𝐺 = 𝐾))
 
1-Oct-2024funcf2lem 46187 A utility theorem for proving equivalence of "is a functor". (Contributed by Zhi Wang, 1-Oct-2024.)
(𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
 
1-Oct-2024map0cor 46070 A function exists iff an empty codomain is accompanied with an empty domain. (Contributed by Zhi Wang, 1-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ∃𝑓 𝑓:𝐴⟶𝐵))
 
1-Oct-2024f002 46069 A function with an empty codomain must have empty domain. (Contributed by Zhi Wang, 1-Oct-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))
 
1-Oct-2024f1mo 46068 A function that maps a set with at most one element to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
((∃*𝑥 𝑥 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
 
1-Oct-2024f102g 46067 A function that maps the empty set to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
 
1-Oct-2024f1sn2g 46066 A function that maps a singleton to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → 𝐹:{𝐴}–1-1→𝐵)
 
1-Oct-2024fdomne0 46065 A function with non-empty domain is non-empty and has non-empty codomain. (Contributed by Zhi Wang, 1-Oct-2024.)
((𝐹:𝑋⟶𝑌 ∧ 𝑋 ≠ ∅) → (𝐹 ≠ ∅ ∧ 𝑌 ≠ ∅))
 
1-Oct-2024mofeu 46063 The uniqueness of a function into a set with at most one element. (Contributed by Zhi Wang, 1-Oct-2024.)
𝐺 = (𝐴 × 𝐵)    &   (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))    &   (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)    ⇒   (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
 
1-Oct-2024sticksstones8 40037 Establish mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 1-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐹:𝐴⟶𝐵)
 
1-Oct-2024sticksstones7 40036 Closure property of sticks and stones function. (Contributed by metakunt, 1-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   (𝜑 → 𝐺:(1...(𝐾 + 1))⟶ℕ0)    &   (𝜑 → 𝑋 ∈ (1...𝐾))    &   𝐹 = (𝑥 ∈ (1...𝐾) ↦ (𝑥 + Σ𝑖 ∈ (1...𝑥)(𝐺‘𝑖)))    &   (𝜑 → Σ𝑖 ∈ (1...(𝐾 + 1))(𝐺‘𝑖) = 𝑁)    ⇒   (𝜑 → (𝐹‘𝑋) ∈ (1...(𝑁 + 𝐾)))
 
1-Oct-2024sticksstones6 40035 Function induces an order isomorphism for sticks and stones theorem. (Contributed by metakunt, 1-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   (𝜑 → 𝐺:(1...(𝐾 + 1))⟶ℕ0)    &   (𝜑 → 𝑋 ∈ (1...𝐾))    &   (𝜑 → 𝑌 ∈ (1...𝐾))    &   (𝜑 → 𝑋 < 𝑌)    &   𝐹 = (𝑥 ∈ (1...𝐾) ↦ (𝑥 + Σ𝑖 ∈ (1...𝑥)(𝐺‘𝑖)))    ⇒   (𝜑 → (𝐹‘𝑋) < (𝐹‘𝑌))
 
30-Sep-2024subthinc 46209 A subcategory of a thin category is thin. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐷 = (𝐶 ↾cat 𝐽)    &   (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   (𝜑 → 𝐶 ∈ ThinCat)    ⇒   (𝜑 → 𝐷 ∈ ThinCat)
 
30-Sep-2024topdlat 46178 A topology is a distributive lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    ⇒   (𝐽 ∈ Top → 𝐼 ∈ DLat)
 
30-Sep-2024toplatmeet 46177 Meets in a topology are realized by intersections. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    &   (𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝐴 ∈ 𝐽)    &   (𝜑 → 𝐵 ∈ 𝐽)    &    ∧ = (meet‘𝐼)    ⇒   (𝜑 → (𝐴 ∧ 𝐵) = (𝐴 ∩ 𝐵))
 
30-Sep-2024toplatjoin 46176 Joins in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    &   (𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝐴 ∈ 𝐽)    &   (𝜑 → 𝐵 ∈ 𝐽)    &    ∨ = (join‘𝐼)    ⇒   (𝜑 → (𝐴 ∨ 𝐵) = (𝐴 ∪ 𝐵))
 
30-Sep-2024toplatglb 46175 Greatest lower bounds in a topology are realized by the interior of the intersection. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    &   (𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ 𝐽)    &   𝐺 = (glb‘𝐼)    &   (𝜑 → 𝑆 ≠ ∅)    ⇒   (𝜑 → (𝐺‘𝑆) = ((int‘𝐽)‘∩ 𝑆))
 
30-Sep-2024toplatlub 46174 Least upper bounds in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    &   (𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ 𝐽)    &   𝑈 = (lub‘𝐼)    ⇒   (𝜑 → (𝑈‘𝑆) = ∪ 𝑆)
 
30-Sep-2024toplatglb0 46173 The empty intersection in a topology is realized by the base set. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    &   (𝜑 → 𝐽 ∈ Top)    &   𝐺 = (glb‘𝐼)    ⇒   (𝜑 → (𝐺‘∅) = ∪ 𝐽)
 
30-Sep-2024topclat 46172 A topology is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    ⇒   (𝐽 ∈ Top → 𝐼 ∈ CLat)
 
30-Sep-2024mreclat 46171 A Moore space is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐶)    ⇒   (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat)
 
30-Sep-2024ipoglb0 46168 The GLB of the empty set is the union of the base. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐺 = (glb‘𝐼))    &   (𝜑 → ∪ 𝐹 ∈ 𝐹)    ⇒   (𝜑 → (𝐺‘∅) = ∪ 𝐹)
 
30-Sep-2024ipolub00 46167 The LUB of the empty set is the empty set if it is contained. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝑈 = (lub‘𝐼))    &   (𝜑 → ∅ ∈ 𝐹)    ⇒   (𝜑 → (𝑈‘∅) = ∅)
 
30-Sep-2024ipolub0 46166 The LUB of the empty set is the intersection of the base. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝑈 = (lub‘𝐼))    &   (𝜑 → ∩ 𝐹 ∈ 𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    ⇒   (𝜑 → (𝑈‘∅) = ∩ 𝐹)
 
30-Sep-2024cbvriotavw 7222 Change bound variable in a restricted description binder. Version of cbvriotav 7227 with a disjoint variable condition, which requires fewer axioms . (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
 
30-Sep-2024cbviotavw 6384 Change bound variables in a description binder. Version of cbviotav 6387 with a disjoint variable condition, which requires fewer axioms . (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (℩𝑥𝜑) = (℩𝑦𝜓)
 
30-Sep-2024rexprg 4629 Convert a restricted existential quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2139, ax-12 2173. (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)))
 
30-Sep-2024ralprg 4627 Convert a restricted universal quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2139, ax-12 2173. (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))
 
30-Sep-2024rexsng 4607 Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) Avoid ax-10 2139, ax-12 2173. (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
 
30-Sep-2024ralsng 4606 Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2139, ax-12 2173. (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
 
30-Sep-2024ralidmw 4435 Idempotent law for restricted quantifier. Weak version of ralidm 4439, which does not require ax-10 2139, ax-12 2173, but requires ax-8 2110. (Contributed by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
 
30-Sep-2024rabeq0w 4314 Condition for a restricted class abstraction to be empty. Version of rabeq0 4315 using implicit substitution, which does not require ax-10 2139, ax-11 2156, ax-12 2173, but requires ax-8 2110. (Contributed by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
 
30-Sep-2024cbvreuvw 3375 Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3379 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
 
30-Sep-2024cbvrmovw 3374 Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmov 3380 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
 
30-Sep-2024cbveuvw 2606 Change bound variable. Uses only Tarski's FOL axiom schemes. See cbveu 2609 for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 25-Nov-1994.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
 
30-Sep-2024cbvmovw 2602 Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvmo 2605 and cbvmow 2603 for versions with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
 
29-Sep-2024oppcthin 46208 The opposite category of a thin category is thin. (Contributed by Zhi Wang, 29-Sep-2024.)
𝑂 = (oppCat‘𝐶)    ⇒   (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat)
 
29-Sep-2024mrelatglbALT 46170 Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐼 = (toInc‘𝐶)    &   𝐺 = (glb‘𝐼)    ⇒   ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈)
 
29-Sep-2024mrelatlubALT 46169 Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐼 = (toInc‘𝐶)    &   𝐹 = (mrCls‘𝐶)    &   𝐿 = (lub‘𝐼)    ⇒   ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))
 
29-Sep-2024ipoglb 46165 The GLB of the inclusion poset. (hypotheses "ipolub.s" and "ipoglb.t" could be eliminated with 𝑆 ∈ dom 𝐺.) Could be significantly shortened if posglbdg 18048 is in quantified form. mrelatglb 18193 could potentially be shortened using this. See mrelatglbALT 46170. (Contributed by Zhi Wang, 29-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐹)    &   (𝜑 → 𝐺 = (glb‘𝐼))    &   (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆})    &   (𝜑 → 𝑇 ∈ 𝐹)    ⇒   (𝜑 → (𝐺‘𝑆) = 𝑇)
 
29-Sep-2024ipoglbdm 46164 The domain of the GLB of the inclusion poset. (Contributed by Zhi Wang, 29-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐹)    &   (𝜑 → 𝐺 = (glb‘𝐼))    &   (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆})    ⇒   (𝜑 → (𝑆 ∈ dom 𝐺 ↔ 𝑇 ∈ 𝐹))
 
29-Sep-2024ipoglblem 46163 Lemma for ipoglbdm 46164 and ipoglb 46165. (Contributed by Zhi Wang, 29-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐹)    &    ≤ = (le‘𝐼)    ⇒   ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((𝑋 ⊆ ∩ 𝑆 ∧ ∀𝑧 ∈ 𝐹 (𝑧 ⊆ ∩ 𝑆 → 𝑧 ⊆ 𝑋)) ↔ (∀𝑦 ∈ 𝑆 𝑋 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑋))))
 
29-Sep-2024unilbeu 46159 Existential uniqueness of the greatest lower bound. (Contributed by Zhi Wang, 29-Sep-2024.)
(𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) ↔ 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}))
 
29-Sep-2024isclatd 46157 The predicate "is a complete lattice" (deduction form). (Contributed by Zhi Wang, 29-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝑈 = (lub‘𝐾))    &   (𝜑 → 𝐺 = (glb‘𝐾))    &   (𝜑 → 𝐾 ∈ Poset)    &   ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝑈)    &   ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝐺)    ⇒   (𝜑 → 𝐾 ∈ CLat)
 
29-Sep-2024glbeldm2d 46141 Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 29-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → ≤ = (le‘𝐾))    &   (𝜑 → 𝐺 = (glb‘𝐾))    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))    &   (𝜑 → 𝐾 ∈ Poset)    ⇒   (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
 
29-Sep-2024mreuniss 46081 The union of a collection of closed sets is a subset. (Contributed by Zhi Wang, 29-Sep-2024.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → ∪ 𝑆 ⊆ 𝑋)
 
29-Sep-2024unilbss 46051 Superclass of the greatest lower bound. A dual statement of ssintub 4894. (Contributed by Zhi Wang, 29-Sep-2024.)
∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴
 
29-Sep-2024focofob 44459 If the domain of a function 𝐺 equals the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 and 𝐹 as function to the domain of 𝐺 are both surjective. Symmetric version of fnfocofob 44458 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 29-Sep-2024.)
((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ (𝐹:𝐴–onto→𝐶 ∧ 𝐺:𝐶–onto→𝐷)))
 
29-Sep-2024fnfocofob 44458 If the domain of a function 𝐺 equals the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 is surjective. (Contributed by GL and AV, 29-Sep-2024.)
((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))
 
29-Sep-2024funfocofob 44457 If the domain of a function 𝐺 is a subset of the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 is surjective. (Contributed by GL and AV, 29-Sep-2024.)
((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))
 
29-Sep-2024fresfo 44429 Conditions for a restriction to be an onto function. Part of fresf1o 30867. (Contributed by AV, 29-Sep-2024.)
((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–onto→𝐶)
 
29-Sep-2024f1imaenfi 8939 If a function is one-to-one, then the image of a finite subset of its domain under it is equinumerous to the subset. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1imaeng 8755). (Contributed by BTernaryTau, 29-Sep-2024.)
((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ Fin) → (𝐹 “ 𝐶) ≈ 𝐶)
 
29-Sep-2024rescnvimafod 6933 The restriction of a function to a preimage of a class is a function onto the intersection of this class and the range of the function. (Contributed by AV, 13-Sep-2024.) (Revised by AV, 29-Sep-2024.)
(𝜑 → 𝐹 Fn 𝐴)    &   (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵))    &   (𝜑 → 𝐷 = (◡𝐹 “ 𝐵))    ⇒   (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸)
 
29-Sep-2024foco 6686 Composition of onto functions. (Contributed by NM, 22-Mar-2006.) (Proof shortened by AV, 29-Sep-2024.)
((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶)
 
29-Sep-2024focofo 6685 Composition of onto functions. Generalisation of foco 6686. (Contributed by AV, 29-Sep-2024.)
((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵)
 
29-Sep-2024focnvimacdmdm 6684 The preimage of the codomain of a surjection is its domain. (Contributed by AV, 29-Sep-2024.)
(𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴)
 
28-Sep-2024ipolub 46162 The LUB of the inclusion poset. (hypotheses "ipolub.s" and "ipolub.t" could be eliminated with 𝑆 ∈ dom 𝑈.) Could be significantly shortened if poslubdg 18047 is in quantified form. mrelatlub 18195 could potentially be shortened using this. See mrelatlubALT 46169. (Contributed by Zhi Wang, 28-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐹)    &   (𝜑 → 𝑈 = (lub‘𝐼))    &   (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥})    &   (𝜑 → 𝑇 ∈ 𝐹)    ⇒   (𝜑 → (𝑈‘𝑆) = 𝑇)
 
28-Sep-2024ipolubdm 46161 The domain of the LUB of the inclusion poset. (Contributed by Zhi Wang, 28-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐹)    &   (𝜑 → 𝑈 = (lub‘𝐼))    &   (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥})    ⇒   (𝜑 → (𝑆 ∈ dom 𝑈 ↔ 𝑇 ∈ 𝐹))
 
28-Sep-2024ipolublem 46160 Lemma for ipolubdm 46161 and ipolub 46162. (Contributed by Zhi Wang, 28-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐹)    &    ≤ = (le‘𝐼)    ⇒   ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((∪ 𝑆 ⊆ 𝑋 ∧ ∀𝑧 ∈ 𝐹 (∪ 𝑆 ⊆ 𝑧 → 𝑋 ⊆ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑋 ≤ 𝑧))))
 
28-Sep-2024intubeu 46158 Existential uniqueness of the least upper bound. (Contributed by Zhi Wang, 28-Sep-2024.)
(𝐶 ∈ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) ↔ 𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}))
 
28-Sep-2024lubeldm2d 46140 Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 28-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → ≤ = (le‘𝐾))    &   (𝜑 → 𝑈 = (lub‘𝐾))    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))    &   (𝜑 → 𝐾 ∈ Poset)    ⇒   (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
 
28-Sep-2024rspceb2dv 46036 Restricted existential specialization, using implicit substitution in both directions. (Contributed by Zhi Wang, 28-Sep-2024.)
((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 → 𝜒))    &   ((𝜑 ∧ 𝜒) → 𝐴 ∈ 𝐵)    &   ((𝜑 ∧ 𝜒) → 𝜃)    &   (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒))
 
28-Sep-2024sticksstones5 40034 Count the number of strictly monotonely increasing functions on finite domains and codomains. (Contributed by metakunt, 28-Sep-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → (♯‘𝐴) = (𝑁C𝐾))
 
28-Sep-2024sticksstones4 40033 Equinumerosity lemma for sticks and stones. (Contributed by metakunt, 28-Sep-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐴 ≈ 𝐵)
 
28-Sep-2024sticksstones3 40032 The range function on strictly monotone functions with finite domain and codomain is an surjective mapping onto 𝐾-elemental sets. (Contributed by metakunt, 28-Sep-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    &   𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧)    ⇒   (𝜑 → 𝐹:𝐴–onto→𝐵)
 
27-Sep-2024posmidm 46155 Poset meet is idempotent. latmidm 18107 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.)
𝐵 = (Base‘𝐾)    &    ∧ = (meet‘𝐾)    ⇒   ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) = 𝑋)
 
27-Sep-2024posjidm 46154 Poset join is idempotent. latjidm 18095 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.)
𝐵 = (Base‘𝐾)    &    ∨ = (join‘𝐾)    ⇒   ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = 𝑋)
 
27-Sep-2024inpw 46052 Two ways of expressing a collection of subsets as seen in df-ntr 22079, unimax 4874, and others (Contributed by Zhi Wang, 27-Sep-2024.)
(𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})
 
27-Sep-2024mpbiran4d 46031 Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 27-Sep-2024.)
(𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    &   ((𝜑 ∧ 𝜃) → 𝜒)    ⇒   (𝜑 → (𝜓 ↔ 𝜃))
 
27-Sep-2024sticksstones2 40031 The range function on strictly monotone functions with finite domain and codomain is an injective mapping onto 𝐾-elemental sets. (Contributed by metakunt, 27-Sep-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    &   𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧)    ⇒   (𝜑 → 𝐹:𝐴–1-1→𝐵)
 
27-Sep-2024sticksstones1 40030 Different strictly monotone functions have different ranges. (Contributed by metakunt, 27-Sep-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   𝐼 = inf({𝑧 ∈ (1...𝐾) ∣ (𝑋‘𝑧) ≠ (𝑌‘𝑧)}, ℝ, < )    ⇒   (𝜑 → ran 𝑋 ≠ ran 𝑌)
 
27-Sep-2024cofcutrtime 34020 If 𝑋 is the cut of 𝐴 and 𝐵 and all of 𝐴 and 𝐵 are older than 𝑋, then ( L ‘𝑋) is cofinal with 𝐴 and ( R ‘𝑋) is coinitial with 𝐵. Note: we will call a cut where all of the elements of the cut are older than the cut itself a "timely" cut. Part of Theorem 4.02(12) of [Alling] p. 125. (Contributed by Scott Fenton, 27-Sep-2024.)
(((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧))
 
27-Sep-2024imaeqsalv 33594 Substitute a function value into a universal quantifier over an image. (Contributed by Scott Fenton, 27-Sep-2024.)
(𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))
 
27-Sep-2024imaeqsexv 33593 Substitute a function value into an existential quantifier over an image. (Contributed by Scott Fenton, 27-Sep-2024.)
(𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓))
 
27-Sep-2024rlimcn3 15227 Image of a limit under a continuous map, two-arg version. Originally a subproof of rlimcn2 15228. (Contributed by SN, 27-Sep-2024.)
((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝑋)    &   ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐶 ∈ 𝑌)    &   ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐵𝐹𝐶) ∈ ℂ)    &   (𝜑 → (𝑅𝐹𝑆) ∈ ℂ)    &   (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝑅)    &   (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((abs‘(𝑢 − 𝑅)) < 𝑟 ∧ (abs‘(𝑣 − 𝑆)) < 𝑠) → (abs‘((𝑢𝐹𝑣) − (𝑅𝐹𝑆))) < 𝑥))    ⇒   (𝜑 → (𝑧 ∈ 𝐴 ↦ (𝐵𝐹𝐶)) ⇝𝑟 (𝑅𝐹𝑆))
 
26-Sep-2024postcpos 46247 The converted category is a poset iff the original proset is a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    ⇒   (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset))
 
26-Sep-2024toslat 46156 A toset is a lattice. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝐾 ∈ Toset → 𝐾 ∈ Lat)
 
26-Sep-2024glbpr 46149 The GLB of the set of two comparable elements in a poset is the less one of the two. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   𝐵 = (Base‘𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    ≤ = (le‘𝐾)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑆 = {𝑋, 𝑌})    &   𝐺 = (glb‘𝐾)    ⇒   (𝜑 → (𝐺‘𝑆) = 𝑋)
 
26-Sep-2024glbprdm 46148 The set of two comparable elements in a poset has GLB. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   𝐵 = (Base‘𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    ≤ = (le‘𝐾)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑆 = {𝑋, 𝑌})    &   𝐺 = (glb‘𝐾)    ⇒   (𝜑 → 𝑆 ∈ dom 𝐺)
 
26-Sep-2024glbprlem 46147 Lemma for glbprdm 46148 and glbpr 46149. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   𝐵 = (Base‘𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    ≤ = (le‘𝐾)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑆 = {𝑋, 𝑌})    &   𝐺 = (glb‘𝐾)    ⇒   (𝜑 → (𝑆 ∈ dom 𝐺 ∧ (𝐺‘𝑆) = 𝑋))
 
26-Sep-2024lubpr 46146 The LUB of the set of two comparable elements in a poset is the greater one of the two. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   𝐵 = (Base‘𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    ≤ = (le‘𝐾)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑆 = {𝑋, 𝑌})    &   𝑈 = (lub‘𝐾)    ⇒   (𝜑 → (𝑈‘𝑆) = 𝑌)
 
26-Sep-2024lubprdm 46145 The set of two comparable elements in a poset has LUB. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   𝐵 = (Base‘𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    ≤ = (le‘𝐾)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑆 = {𝑋, 𝑌})    &   𝑈 = (lub‘𝐾)    ⇒   (𝜑 → 𝑆 ∈ dom 𝑈)
 
26-Sep-2024lubprlem 46144 Lemma for lubprdm 46145 and lubpr 46146. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   𝐵 = (Base‘𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    ≤ = (le‘𝐾)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑆 = {𝑋, 𝑌})    &   𝑈 = (lub‘𝐾)    ⇒   (𝜑 → (𝑆 ∈ dom 𝑈 ∧ (𝑈‘𝑆) = 𝑌))
 
26-Sep-2024glbsscl 46143 If a subset of 𝑆 contains the GLB of 𝑆, then the two sets have the same GLB. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   (𝜑 → 𝑇 ⊆ 𝑆)    &   𝐺 = (glb‘𝐾)    &   (𝜑 → 𝑆 ∈ dom 𝐺)    &   (𝜑 → (𝐺‘𝑆) ∈ 𝑇)    ⇒   (𝜑 → (𝑇 ∈ dom 𝐺 ∧ (𝐺‘𝑇) = (𝐺‘𝑆)))
 
26-Sep-2024lubsscl 46142 If a subset of 𝑆 contains the LUB of 𝑆, then the two sets have the same LUB. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   (𝜑 → 𝑇 ⊆ 𝑆)    &   𝑈 = (lub‘𝐾)    &   (𝜑 → 𝑆 ∈ dom 𝑈)    &   (𝜑 → (𝑈‘𝑆) ∈ 𝑇)    ⇒   (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈‘𝑇) = (𝑈‘𝑆)))
 
26-Sep-2024glbeldm2 46139 Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
𝐵 = (Base‘𝐾)    &    ≤ = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))    &   (𝜑 → 𝐾 ∈ Poset)    ⇒   (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
 
26-Sep-2024lubeldm2 46138 Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
𝐵 = (Base‘𝐾)    &    ≤ = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))    &   (𝜑 → 𝐾 ∈ Poset)    ⇒   (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
 
26-Sep-2024sn-wcdeq 40097 Alternative to wcdeq 3693 and df-cdeq 3694. This flattens the syntax representation ( wi ( weq vx vy ) wph ) to ( sn-wcdeq vx vy wph ), illustrating the comment of df-cdeq 3694. (Contributed by SN, 26-Sep-2024.) (New usage is discouraged.)
wff (𝑥 = 𝑦 → 𝜑)
 
26-Sep-2024fvpr2 7049 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by BJ, 26-Sep-2024.)
𝐵 ∈ V    &   𝐷 ∈ V    ⇒   (𝐴 ≠ 𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)
 
26-Sep-2024fvpr1 7047 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by BJ, 26-Sep-2024.)
𝐴 ∈ V    &   𝐶 ∈ V    ⇒   (𝐴 ≠ 𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)
 
26-Sep-2024fvpr2g 7045 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Proof shortened by BJ, 26-Sep-2024.)
((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)
 
26-Sep-202419.8aw 2054 If a formula is true, then it is true for at least one instance. This is to 19.8a 2176 what spw 2038 is to sp 2178. (Contributed by SN, 26-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (𝜑 → ∃𝑥𝜑)
 
25-Sep-2024postc 46249 The converted category is a poset iff no distinct objects are isomorphic. (Contributed by Zhi Wang, 25-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   𝐵 = (Base‘𝐶)    ⇒   (𝜑 → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥( ≃𝑐 ‘𝐶)𝑦 → 𝑥 = 𝑦)))
 
25-Sep-2024postcposALT 46248 Alternate proof for postcpos 46247. (Contributed by Zhi Wang, 25-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    ⇒   (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset))
 
25-Sep-2024thinccic 46230 In a thin category, two objects are isomorphic iff there are morphisms between them in both directions. (Contributed by Zhi Wang, 25-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅)))
 
25-Sep-2024thinciso 46229 In a thin category, 𝐹:𝑋⟶𝑌 is an isomorphism iff there is a morphism from 𝑌 to 𝑋. (Contributed by Zhi Wang, 25-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Iso‘𝐶)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝑌𝐻𝑋) ≠ ∅))
 
25-Sep-2024endmndlem 46184 A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 46251 for converting a monoid to a category. Lemma for bj-endmnd 35416. (Contributed by Zhi Wang, 25-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → (𝑋𝐻𝑋) = (Base‘𝑀))    &   (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (+g‘𝑀))    ⇒   (𝜑 → 𝑀 ∈ Mnd)
 
25-Sep-2024meetdm3 46153 The meet of any two elements always exists iff all unordered pairs have GLB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.)
𝐵 = (Base‘𝐾)    &   (𝜑 → 𝐾 ∈ 𝑉)    &   𝐺 = (glb‘𝐾)    &    ∧ = (meet‘𝐾)    &    ≤ = (le‘𝐾)    ⇒   (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))
 
25-Sep-2024meetdm2 46152 The meet of any two elements always exists iff all unordered pairs have GLB. (Contributed by Zhi Wang, 25-Sep-2024.)
𝐵 = (Base‘𝐾)    &   (𝜑 → 𝐾 ∈ 𝑉)    &   𝐺 = (glb‘𝐾)    &    ∧ = (meet‘𝐾)    ⇒   (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
 
25-Sep-2024joindm3 46151 The join of any two elements always exists iff all unordered pairs have LUB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.)
𝐵 = (Base‘𝐾)    &   (𝜑 → 𝐾 ∈ 𝑉)    &   𝑈 = (lub‘𝐾)    &    ∨ = (join‘𝐾)    &    ≤ = (le‘𝐾)    ⇒   (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))
 
25-Sep-2024joindm2 46150 The join of any two elements always exists iff all unordered pairs have LUB. (Contributed by Zhi Wang, 25-Sep-2024.)
𝐵 = (Base‘𝐾)    &   (𝜑 → 𝐾 ∈ 𝑉)    &   𝑈 = (lub‘𝐾)    &    ∨ = (join‘𝐾)    ⇒   (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈))
 
25-Sep-2024cofcutr 34019 If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐴 is cofinal with ( L ‘𝑋) and 𝐵 is coinitial with ( R ‘𝑋). Theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 25-Sep-2024.)
((𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ ( L ‘𝑋)∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ ( R ‘𝑋)∃𝑤 ∈ 𝐵 𝑤 ≤s 𝑧))
 
25-Sep-2024cofcut2 34018 If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)
(((𝐴 <<s 𝐵 ∧ 𝐶 ∈ 𝒫 No ∧ 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
 
25-Sep-2024cofcut1 34017 If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷. Then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)
((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
 
25-Sep-2024ssltsepcd 33915 Two elements of separated sets obey less than. Deduction form of ssltsepc 33914. (Contributed by Scott Fenton, 25-Sep-2024.)
(𝜑 → 𝐴 <<s 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → 𝑋 <s 𝑌)
 
25-Sep-2024f1domfi 8928 If the codomain of a one-to-one function is finite, then the function's domain is dominated by its codomain. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1domg 8715). (Contributed by BTernaryTau, 25-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
 
25-Sep-2024en2sn 8785 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) Avoid ax-pow 5283. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7566. (Revised by BTernaryTau, 25-Sep-2024.)
((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵})
 
25-Sep-2024f1dom2g 8712 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 8715 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.) (Proof shortened by BTernaryTau, 25-Sep-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
 
25-Sep-2024rexeqbidvv 3330 Version of rexeqbidv 3328 with additional disjoint variable conditions, not requiring ax-8 2110 nor df-clel 2817. (Contributed by Wolf Lammen, 25-Sep-2024.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
 
25-Sep-2024nfnaew 2147 All variables are effectively bound in a distinct variable specifier. Version of nfnae 2434 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 25-Sep-2024.)
Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
 
24-Sep-2024mndtcbas2 46256 Two objects in a category built from a monoid are identical. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → 𝑋 = 𝑌)
 
24-Sep-2024thincinv 46228 In a thin category, 𝐹 is an inverse of 𝐺 iff 𝐹 is a section of 𝐺 (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑆 = (Sect‘𝐶)    &   𝑁 = (Inv‘𝐶)    ⇒   (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐹(𝑋𝑆𝑌)𝐺))
 
24-Sep-2024thincsect2 46227 In a thin category, 𝐹 is a section of 𝐺 iff 𝐺 is a section of 𝐹. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑆 = (Sect‘𝐶)    ⇒   (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐺(𝑌𝑆𝑋)𝐹))
 
24-Sep-2024thincsect 46226 In a thin category, one morphism is a section of another iff they are pointing towards each other. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑆 = (Sect‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))))
 
24-Sep-2024thincepi 46204 In a thin category, all morphisms are epimorphisms. The converse does not hold. See grptcepi 46264. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐸 = (Epi‘𝐶)    ⇒   (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))
 
24-Sep-2024thincmon 46203 In a thin category, all morphisms are monomorphisms. The converse does not hold. See grptcmon 46263. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑀 = (Mono‘𝐶)    ⇒   (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))
 
24-Sep-2024thincid 46202 In a thin category, a morphism from an object to itself is an identity morphism. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &    1 = (Id‘𝐶)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑋))    ⇒   (𝜑 → 𝐹 = ( 1 ‘𝑋))
 
24-Sep-2024thinccd 46194 A thin category is a category (deduction form). (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    ⇒   (𝜑 → 𝐶 ∈ Cat)
 
24-Sep-2024mofsssn 46061 There is at most one function into a subclass of a singleton. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝐵 ⊆ {𝑌} → ∃*𝑓 𝑓:𝐴⟶𝐵)
 
24-Sep-2024mpbiran3d 46030 Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    &   ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   (𝜑 → (𝜓 ↔ 𝜒))
 
24-Sep-2024coinitsslt 34016 If 𝐵 is coinitial with 𝐶 and 𝐴 precedes 𝐶, then 𝐴 precedes 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)
((𝐵 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s 𝐵)
 
24-Sep-2024cofsslt 34015 If every element of 𝐴 is bounded by some element of 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Note - we will often use the term "cofinal" to denote that every element of 𝐴 is bounded above by some element of 𝐵. Similarly, we will use the term "coinitial" to denote that every element of 𝐴 is bounded below by some element of 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)
((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)
 
24-Sep-2024ssltd 33913 Deduce surreal set less than. (Contributed by Scott Fenton, 24-Sep-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐴 ⊆ No )    &   (𝜑 → 𝐵 ⊆ No )    &   ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 <s 𝑦)    ⇒   (𝜑 → 𝐴 <<s 𝐵)
 
24-Sep-2024setc2ohom 17726 (SetCat‘2o) is a category (provable from setccat 17716 and 2oex 8284) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 17725. Notably, the empty set ∅ is simultaneously an object (setc2obas 17725) , an identity morphism from ∅ to ∅ (setcid 17717 or thincid 46202) , and a non-identity morphism from ∅ to 1o. See cat1lem 17727 and cat1 17728 for a more general statement. This category is also thin (setc2othin 46225), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 46223 for more details on the "equivalence". (Contributed by Zhi Wang, 24-Sep-2024.)
𝐶 = (SetCat‘2o)    &   𝐻 = (Hom ‘𝐶)    ⇒   ∅ ∈ ((∅𝐻∅) ∩ (∅𝐻1o))
 
24-Sep-2024setc2obas 17725 ∅ and 1o are distinct objects in (SetCat‘2o). This combined with setc2ohom 17726 demonstrates that the category does not have pairwise disjoint hom-sets. See also df-cat 17294 and cat1 17728. (Contributed by Zhi Wang, 24-Sep-2024.)
𝐶 = (SetCat‘2o)    &   𝐵 = (Base‘𝐶)    ⇒   (∅ ∈ 𝐵 ∧ 1o ∈ 𝐵 ∧ 1o ≠ ∅)
 
24-Sep-2024en1uniel 8772 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7566. (Revised by BTernaryTau, 24-Sep-2024.)
(𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)
 
24-Sep-2024en1b 8767 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7566. (Revised by BTernaryTau, 24-Sep-2024.)
(𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})
 
24-Sep-2024eqsnuniex 5278 If a class is equal to the singleton of its union, then its union exists. (Contributed by BTernaryTau, 24-Sep-2024.)
(𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V)
 
24-Sep-2024nfra2wOLD 3152 Obsolete version of nfra2w 3151 as of 31-Oct-2024. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by Gino Giotto, 24-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
 
24-Sep-2024nfraldw 3146 Deduction version of nfralw 3149. Version of nfrald 3148 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 24-Sep-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)
 
23-Sep-2024grptcepi 46264 All morphisms in a category converted from a group are epimorphisms. (Contributed by Zhi Wang, 23-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝐺))    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝐸 = (Epi‘𝐶))    ⇒   (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))
 
23-Sep-2024grptcmon 46263 All morphisms in a category converted from a group are monomorphisms. (Contributed by Zhi Wang, 23-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝐺))    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝑀 = (Mono‘𝐶))    ⇒   (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))
 
23-Sep-2024catprsc2 46183 An alternate construction of the preorder induced by a category. See catprs2 46181 for details. See also catprsc 46182 for a different construction. The two constructions are different because df-cat 17294 does not require the domain of 𝐻 to be 𝐵 × 𝐵. (Contributed by Zhi Wang, 23-Sep-2024.)
(𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅})    ⇒   (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
 
23-Sep-2024mosssn2 46050 Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 23-Sep-2024.)
(∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})
 
23-Sep-2024mosssn 46048 "At most one" element in a subclass of a singleton. (Contributed by Zhi Wang, 23-Sep-2024.)
(𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
 
23-Sep-2024reutruALT 46040 Alternate proof for reutru 46039. (Contributed by Zhi Wang, 23-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
 
23-Sep-2024reutru 46039 Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
(∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
 
23-Sep-2024rmotru 46038 Two ways of expressing "at most one" element. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by BJ, 23-Sep-2024.)
(∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤)
 
23-Sep-2024rextru 46037 Two ways of expressing "at least one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
(∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤)
 
23-Sep-2024dtrucor3 46032 An example of how ax-5 1914 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5288 in the ZF set theory. axc16nf 2258 and euae 2661 demonstrate that the violation of dtru 5288 leads to a model with only one object assuming its existence (ax-6 1972). The conclusion is also provable in the empty model ( see emptyal 1912). See also nf5 2282 and nf5i 2144 for the relation between unconditional ax-5 1914 and being not free. (Contributed by Zhi Wang, 23-Sep-2024.)
¬ ∀𝑥 𝑥 = 𝑦    &   (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)    ⇒   ∀𝑥 𝑥 = 𝑦
 
23-Sep-2024entrfir 8937 Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8747). (Contributed by BTernaryTau, 23-Sep-2024.)
((𝐶 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
 
23-Sep-2024entrfi 8936 Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8747). (Contributed by BTernaryTau, 23-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
 
23-Sep-2024enfi 8933 Equinumerous sets have the same finiteness. For a shorter proof using ax-pow 5283, see enfiALT 8934. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5283. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
 
23-Sep-2024enfii 8932 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5283. (Revised by BTernaryTau, 23-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
 
23-Sep-2024ssnnfi 8914 A subset of a natural number is finite. (Contributed by NM, 24-Jun-1998.) (Proof shortened by BTernaryTau, 23-Sep-2024.)
((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
 
23-Sep-2024nnfi 8912 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.) Avoid ax-pow 5283. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ∈ ω → 𝐴 ∈ Fin)
 
23-Sep-2024en1 8765 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7566. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
 
23-Sep-2024ensn1 8761 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7566. (Revised by BTernaryTau, 23-Sep-2024.)
𝐴 ∈ V    ⇒   {𝐴} ≈ 1o
 
23-Sep-2024en0 8758 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5283, ax-un 7566. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ≈ ∅ ↔ 𝐴 = ∅)
 
23-Sep-2024bren 8701 Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) (Proof shortened by BTernaryTau, 23-Sep-2024.)
(𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
 
23-Sep-2024breng 8700 Equinumerosity relation. This variation of bren 8701 does not require the Axiom of Union. (Contributed by BTernaryTau, 23-Sep-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
 
23-Sep-2024noel 4261 The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) Remove dependency on ax-10 2139, ax-11 2156, and ax-12 2173. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
¬ 𝐴 ∈ ∅
 
23-Sep-2024dfnul3 4257 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) (Proof shortened by BJ, 23-Sep-2024.)
∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}
 
23-Sep-2024dfnul2 4256 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2139, ax-11 2156, and ax-12 2173. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
 
23-Sep-2024dfnul4 4255 Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2110, df-clel 2817. (Revised by Gino Giotto, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4256. (Revised by BJ, 23-Sep-2024.)
∅ = {𝑥 ∣ ⊥}
 
23-Sep-2024nfabdw 2929 Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2931 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Sep-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
 
22-Sep-2024mndtcid 46262 The identity morphism, or identity arrow, of the category built from a monoid is the identity element of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 1 = (Id‘𝐶))    ⇒   (𝜑 → ( 1 ‘𝑋) = (0g‘𝑀))
 
22-Sep-2024mndtccat 46261 The function value is a category. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    ⇒   (𝜑 → 𝐶 ∈ Cat)
 
22-Sep-2024mndtccatid 46260 Lemma for mndtccat 46261 and mndtcid 46262. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    ⇒   (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0g‘𝑀))))
 
22-Sep-2024mndtcco2 46259 The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → · = (comp‘𝐶))    &   (𝜑 → ⚬ = (⟨𝑋, 𝑌⟩ · 𝑍))    ⇒   (𝜑 → (𝐺 ⚬ 𝐹) = (𝐺(+g‘𝑀)𝐹))
 
22-Sep-2024mndtcco 46258 The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → · = (comp‘𝐶))    ⇒   (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (+g‘𝑀))
 
22-Sep-2024mndtchom 46257 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.)
(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-Sep-2024mndtcob 46255 Lemma for mndtchom 46257 and mndtcco 46258. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝑋 = 𝑀)
 
22-Sep-2024mndtcbas 46254 The category built from a monoid contains precisely one object. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    ⇒   (𝜑 → ∃!𝑥 𝑥 ∈ 𝐵)
 
22-Sep-2024mndtcbasval 46253 The base set of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    ⇒   (𝜑 → 𝐵 = {𝑀})
 
22-Sep-2024mndtcval 46252 Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    ⇒   (𝜑 → 𝐶 = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})
 
22-Sep-2024df-mndtc 46251 Definition of the function converting a monoid to a category. Example 3.3(4.e) of [Adamek] p. 24.

The definition of the base set is arbitrary. The whole extensible structure becomes the object here (see mndtcbasval 46253) , instead of just the base set, as is the case in Example 3.3(4.e) of [Adamek] p. 24.

The resulting category is defined entirely, up to isomorphism, by mndtcbas 46254, mndtchom 46257, mndtcco 46258. Use those instead.

See example 3.26(3) of [Adamek] p. 33 for more on isomorphism.

"MndToCat" was taken instead of "MndCat" because the latter might mean the category of monoids. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)

MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
 
22-Sep-2024eleq2w2ALT 35147 Alternate proof of eleq2w2 2734 and special instance of eleq2 2827. (Contributed by BJ, 22-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
 
22-Sep-2024bj-nnfeai 34845 Nonfreeness implies the equivalent of ax5ea 1917, inference form. (Contributed by BJ, 22-Sep-2024.)
Ⅎ'𝑥𝜑    ⇒   (∃𝑥𝜑 → ∀𝑥𝜑)
 
22-Sep-2024bj-nnfei 34842 Nonfreeness implies the equivalent of ax5e 1916, inference form. (Contributed by BJ, 22-Sep-2024.)
Ⅎ'𝑥𝜑    ⇒   (∃𝑥𝜑 → 𝜑)
 
22-Sep-2024bj-nnfai 34839 Nonfreeness implies the equivalent of ax-5 1914, inference form. See nf5ri 2191. (Contributed by BJ, 22-Sep-2024.)
Ⅎ'𝑥𝜑    ⇒   (𝜑 → ∀𝑥𝜑)
 
22-Sep-2024cphpyth 24285 The pythagorean theorem for a subcomplex pre-Hilbert space. The square of the norm of the sum of two orthogonal vectors (i.e., whose inner product is 0) is the sum of the squares of their norms. Problem 2 in [Kreyszig] p. 135. This is Metamath 100 proof #4. (Contributed by NM, 17-Apr-2008.) (Revised by SN, 22-Sep-2024.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖‘𝑊)    &    + = (+g‘𝑊)    &   𝑁 = (norm‘𝑊)    &   (𝜑 → 𝑊 ∈ ℂPreHil)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝑁‘(𝐴 + 𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))
 
22-Sep-2024raleqbidvv 3329 Version of raleqbidv 3327 with additional disjoint variable conditions, not requiring ax-8 2110 nor df-clel 2817. (Contributed by BJ, 22-Sep-2024.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
 
22-Sep-2024drnfc2 2927 Formula-building lemma for use with the Distinctor Reduction Theorem. Proof revision is marked as discouraged because the minimizer replaces albidv 1924 with dral2 2438, leading to a one byte longer proof. However feel free to manually edit it according to conventions. (TODO: dral2 2438 depends on ax-13 2372, hence its usage during minimizing is discouraged. Check in the long run whether this is a permanent restriction). Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-8 2110. (Revised by Wolf Lammen, 22-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵))
 
22-Sep-2024drnfc1 2925 Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-8 2110, ax-11 2156. (Revised by Wolf Lammen, 22-Sep-2024.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵))
 
22-Sep-202419.36imv 1949 One direction of 19.36v 1992 that can be proven without ax-6 1972. (Contributed by Rohan Ridenour, 16-Apr-2022.) (Proof shortened by Wolf Lammen, 22-Sep-2024.)
(∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
 
21-Sep-2024prstchom2 46245 Hom-sets of the constructed category are dependent on the preorder.

Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat ( see prstchom2ALT 46246). However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 21-Sep-2024.)

(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   (𝜑 → 𝑌 ∈ (Base‘𝐶))    ⇒   (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
 
21-Sep-2024thincn0eu 46201 In a thin category, a hom-set being non-empty is equivalent to having a unique element. (Contributed by Zhi Wang, 21-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
 
21-Sep-2024thincmod 46200 At most one morphism in each hom-set (deduction form). (Contributed by Zhi Wang, 21-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
 
21-Sep-2024thincmoALT 46199 Alternate proof for thincmo 46198. (Contributed by Zhi Wang, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐶 ∈ ThinCat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
 
21-Sep-2024thincmo 46198 There is at most one morphism in each hom-set. (Contributed by Zhi Wang, 21-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
 
21-Sep-2024idepi 46186 An identity arrow, or an identity morphism, is an epimorphism. (Contributed by Zhi Wang, 21-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   𝐸 = (Epi‘𝐶)    ⇒   (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐸𝑋))
 
21-Sep-2024idmon 46185 An identity arrow, or an identity morphism, is a monomorphism. (Contributed by Zhi Wang, 21-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   𝑀 = (Mono‘𝐶)    ⇒   (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋))
 
21-Sep-2024fineqvacALT 32967 Shorter proof of fineqvac 32966 using ax-rep 5205 and ax-pow 5283. (Contributed by BTernaryTau, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(Fin = V → CHOICE)
 
21-Sep-2024fineqvac 32966 If the Axiom of Infinity is negated, then the Axiom of Choice becomes redundant. For a shorter proof using ax-rep 5205 and ax-pow 5283, see fineqvacALT 32967. (Contributed by BTernaryTau, 21-Sep-2024.)
(Fin = V → CHOICE)
 
21-Sep-2024ffrnb 6599 Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6598. (Contributed by BJ, 21-Sep-2024.)
(𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
 
21-Sep-2024sbalex 2238 Equivalence of two ways to express proper substitution of a setvar for another setvar disjoint from it in a formula. This proof of their equivalence does not use df-sb 2069.

That both sides of the biconditional express proper substitution is proved by sb5 2271 and sb6 2089. The implication "to the left" is equs4v 2004 and does not require ax-10 2139 nor ax-12 2173. It also holds without disjoint variable condition if we allow more axioms (see equs4 2416). Theorem 6.2 of [Quine] p. 40. Theorem equs5 2460 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f 2459 replaces the disjoint variable condition on 𝑥, 𝑡 with the nonfreeness hypothesis of 𝑡 in 𝜑. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2263 in place of equsex 2418 in order to remove dependency on ax-13 2372. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on df-sb 2069. (Revised by BJ, 21-Sep-2024.)

(∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
 
21-Sep-2024cad0 1621 If one input is false, then the adder carry is true exactly when both of the other two inputs are true. (Contributed by Mario Carneiro, 8-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Sep-2024.)
(¬ 𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓)))
 
20-Sep-2024prstchom2ALT 46246 Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 46232. See prstchom2 46245 for a version that does not depend on the definition. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
 
20-Sep-2024prstchom 46244 Hom-sets of the constructed category are dependent on the preorder.

Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat. However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 20-Sep-2024.)

(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   (𝜑 → 𝑌 ∈ (Base‘𝐶))    ⇒   (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
 
20-Sep-2024prstcthin 46243 The preordered set is equipped with a thin category. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    ⇒   (𝜑 → 𝐶 ∈ ThinCat)
 
20-Sep-2024prstcprs 46242 The category is a preordered set. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    ⇒   (𝜑 → 𝐶 ∈ Proset )
 
20-Sep-2024prstchomval 46241 Hom-sets of the constructed category which depend on an arbitrary definition. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐶))    ⇒   (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶))
 
20-Sep-2024prstcoc 46240 Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ⊥ = (oc‘𝐾))    ⇒   (𝜑 → ( ⊥ ‘𝑋) = ((oc‘𝐶)‘𝑋))
 
20-Sep-2024prstcocval 46239 Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ⊥ = (oc‘𝐾))    ⇒   (𝜑 → ⊥ = (oc‘𝐶))
 
20-Sep-2024prstcle 46238 Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐾))    ⇒   (𝜑 → (𝑋 ≤ 𝑌 ↔ 𝑋(le‘𝐶)𝑌))
 
20-Sep-2024prstcleval 46237 Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐾))    ⇒   (𝜑 → ≤ = (le‘𝐶))
 
20-Sep-2024prstcbas 46236 The base set is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → 𝐵 = (Base‘𝐾))    ⇒   (𝜑 → 𝐵 = (Base‘𝐶))
 
20-Sep-2024prstcnid 46235 Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (comp‘ndx)    &   (𝐸‘ndx) ≠ (Hom ‘ndx)    ⇒   (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶))
 
20-Sep-2024prstcnidlem 46234 Lemma for prstcnid 46235 and prstchomval 46241. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (comp‘ndx)    ⇒   (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))
 
20-Sep-2024prstcval 46233 Lemma for prstcnidlem 46234 and prstcthin 46243. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    ⇒   (𝜑 → 𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
 
20-Sep-2024df-prstc 46232 Definition of the function converting a preordered set to a category. Justified by prsthinc 46223.

This definition is somewhat arbitrary. Example 3.3(4.d) of [Adamek] p. 24 demonstrates an alternate definition with pairwise disjoint hom-sets. The behavior of the function is defined entirely, up to isomorphism, by prstcnid 46235, prstchom 46244, and prstcthin 46243. Other important properties include prstcbas 46236, prstcleval 46237, prstcle 46238, prstcocval 46239, prstcoc 46240, prstchom2 46245, and prstcprs 46242. Use those instead.

Note that the defining property prstchom 46244 is equivalent to prstchom2 46245 given prstcthin 46243. See thincn0eu 46201 for justification.

"ProsetToCat" was taken instead of "ProsetCat" because the latter might mean the category of preordered sets (classes). However, "ProsetToCat" seems too long. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)

ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
 
20-Sep-2024setc2othin 46225 The category (SetCat‘2o) is thin. A special case of setcthin 46224. (Contributed by Zhi Wang, 20-Sep-2024.)
(SetCat‘2o) ∈ ThinCat
 
20-Sep-2024setcthin 46224 A category of sets all of whose objects contain at most one element is thin. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (SetCat‘𝑈))    &   (𝜑 → 𝑈 ∈ 𝑉)    &   (𝜑 → ∀𝑥 ∈ 𝑈 ∃*𝑝 𝑝 ∈ 𝑥)    ⇒   (𝜑 → 𝐶 ∈ ThinCat)
 
20-Sep-2024fvconstrn0 46072 Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   (𝜑 → 𝑌 ∈ 𝑉)    &   (𝜑 → 𝑌 ≠ ∅)    ⇒   (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) ≠ ∅))
 
20-Sep-2024mof02 46054 A variant of mof0 46053. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵)
 
20-Sep-2024f1co 6666 Composition of one-to-one functions when the codomain of the first matches the domain of the second. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.) (Proof shortened by AV, 20-Sep-2024.)
((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶)
 
20-Sep-2024funcofd 6617 Composition of two functions as a function with domain and codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof shortened by AV, 20-Sep-2024.)
(𝜑 → Fun 𝐹)    &   (𝜑 → Fun 𝐺)    ⇒   (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)
 
20-Sep-2024fco 6608 Composition of two functions with domain and codomain as a function with domain and codomain. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Proof shortened by AV, 20-Sep-2024.)
((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
 
20-Sep-2024fimacnv 6606 The preimage of the codomain of a function is the function's domain. (Contributed by FL, 25-Jan-2007.) (Proof shortened by AV, 20-Sep-2024.)
(𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴)
 
20-Sep-2024ffrnbd 6600 A function maps to its range iff the the range is a subset of its codomain. Generalization of ffrn 6598. (Contributed by AV, 20-Sep-2024.)
(𝜑 → ran 𝐹 ⊆ 𝐵)    ⇒   (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹))
 
20-Sep-2024fnco 6533 Composition of two functions with domains as a function with domain. (Contributed by NM, 22-May-2006.) (Proof shortened by AV, 20-Sep-2024.)
((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)
 
20-Sep-2024ineqcomi 4134 Two ways of expressing that two classes have a given intersection. Inference form of ineqcom 4133. Disjointness inference when 𝐶 = ∅. (Contributed by Peter Mazsa, 26-Mar-2017.) (Proof shortened by SN, 20-Sep-2024.)
(𝐴 ∩ 𝐵) = 𝐶    ⇒   (𝐵 ∩ 𝐴) = 𝐶
 
20-Sep-2024ecase3ad 1032 Deduction for elimination by cases. (Contributed by NM, 24-May-2013.) (Proof shortened by Wolf Lammen, 20-Sep-2024.)
(𝜑 → (𝜓 → 𝜃))    &   (𝜑 → (𝜒 → 𝜃))    &   (𝜑 → ((¬ 𝜓 ∧ ¬ 𝜒) → 𝜃))    ⇒   (𝜑 → 𝜃)
 
19-Sep-2024indthinc 46221 An indiscrete category in which all hom-sets have exactly one morphism is a thin category. Constructed here is an indiscrete category where all morphisms are ∅. This is a special case of prsthinc 46223, where ≤ = (𝐵 × 𝐵). This theorem also implies a functor from the category of sets to the category of small categories. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof shortened by Zhi Wang, 19-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))    &   (𝜑 → ∅ = (comp‘𝐶))    &   (𝜑 → 𝐶 ∈ 𝑉)    ⇒   (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))
 
19-Sep-2024f1omo 46076 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 46075 assuming ax-un 7566 (see f1omoALT 46077). (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
 
19-Sep-2024mofmo 46062 There is at most one function into a class containing at most one element. (Contributed by Zhi Wang, 19-Sep-2024.)
(∃*𝑥 𝑥 ∈ 𝐵 → ∃*𝑓 𝑓:𝐴⟶𝐵)
 
19-Sep-2024mofsn2 46060 There is at most one function into a singleton. An unconditional variant of mofsn 46059, i.e., the singleton could be empty if 𝑌 is a proper class. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐵 = {𝑌} → ∃*𝑓 𝑓:𝐴⟶𝐵)
 
19-Sep-2024mofsn 46059 There is at most one function into a singleton, with fewer axioms than eufsn 46057 and eufsn2 46058. See also mofsn2 46060. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐵 ∈ 𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})
 
19-Sep-2024eufsn2 46058 There is exactly one function into a singleton, assuming ax-pow 5283 and ax-un 7566. Variant of eufsn 46057. If existence is not needed, use mofsn 46059 or mofsn2 46060 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
 
19-Sep-2024eufsn 46057 There is exactly one function into a singleton, assuming ax-rep 5205. See eufsn2 46058 for different axiom requirements. If existence is not needed, use mofsn 46059 or mofsn2 46060 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
 
19-Sep-2024eufsnlem 46056 There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 46057 assuming ax-rep 5205, or eufsn2 46058 assuming ax-pow 5283 and ax-un 7566. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉)    ⇒   (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
 
19-Sep-2024mof0ALT 46055 Alternate proof for mof0 46053 with stronger requirements on distinct variables. Uses mo4 2566. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
∃*𝑓 𝑓:𝐴⟶∅
 
19-Sep-2024mof0 46053 There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
∃*𝑓 𝑓:𝐴⟶∅
 
19-Sep-2024mo0sn 46049 Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 19-Sep-2024.)
(∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
 
19-Sep-2024mo0 46047 "At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)
 
19-Sep-2024mosn 46046 "At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
 
19-Sep-2024vsn 46045 The singleton of the universal class is the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
{V} = ∅
 
19-Sep-2024f1cof1b 44456 If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is injective iff 𝐹 and 𝐺 are both injective. (Contributed by GL and AV, 19-Sep-2024.)
((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷)))
 
19-Sep-20242oex 8284 2o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2139, ax-11 2156, ax-12 2173, ax-un 7566. (Proof shortened by Zhi Wang, 19-Sep-2024.)
2o ∈ V
 
19-Sep-20241oex 8280 Ordinal 1 is a set. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by AV, 1-Jul-2022.) Remove dependency on ax-10 2139, ax-11 2156, ax-12 2173, ax-un 7566. (Revised by Zhi Wang, 19-Sep-2024.)
1o ∈ V
 
19-Sep-2024ecase2d 1026 Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Sep-2024.)
(𝜑 → 𝜓)    &   (𝜑 → ¬ (𝜓 ∧ 𝜒))    &   (𝜑 → ¬ (𝜓 ∧ 𝜃))    &   (𝜑 → (𝜏 ∨ (𝜒 ∨ 𝜃)))    ⇒   (𝜑 → 𝜏)
 
18-Sep-2024prsthinc 46223 Preordered sets as categories. Similar to example 3.3(4.d) of [Adamek] p. 24, but the hom-sets are not pairwise disjoint. One can define a functor from the category of prosets to the category of small thin categories. See catprs 46180 and catprs2 46181 for inducing a preorder from a category. Example 3.26(2) of [Adamek] p. 33 indicates that it induces a bijection from the equivalence class of isomorphic small thin categories to the equivalence class of order-isomorphic preordered sets. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶))    &   (𝜑 → ∅ = (comp‘𝐶))    &   (𝜑 → ≤ = (le‘𝐶))    &   (𝜑 → 𝐶 ∈ Proset )    ⇒   (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))
 
18-Sep-2024catprsc 46182 A construction of the preorder induced by a category. See catprs2 46181 for details. See also catprsc2 46183 for an alternate construction. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)})    ⇒   (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
 
18-Sep-2024catprs2 46181 A category equipped with the induced preorder, where an object 𝑥 is defined to be "less than or equal to" 𝑦 iff there is a morphism from 𝑥 to 𝑦, is a preordered set, or a proset. The category might not be thin. See catprsc 46182 and catprsc2 46183 for constructions satisfying the hypothesis "catprs.1". See catprs 46180 for a more primitive version. See prsthinc 46223 for constructing a thin category from a proset. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → ≤ = (le‘𝐶))    ⇒   (𝜑 → 𝐶 ∈ Proset )
 
18-Sep-2024catprs 46180 A preorder can be extracted from a category. See catprs2 46181 for more details. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝐶 ∈ Cat)    ⇒   ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))
 
18-Sep-2024catprslem 46179 Lemma for catprs 46180. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
 
18-Sep-2024isprsd 46137 Property of being a preordered set (deduction form). (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → ≤ = (le‘𝐾))    &   (𝜑 → 𝐾 ∈ 𝑉)    ⇒   (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
 
18-Sep-2024f1omoALT 46077 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.) Use f1omo 46076 without assuming ax-un 7566. (Contributed by Zhi Wang, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
 
18-Sep-2024fvconstdomi 46075 A constant function's value is dominated by the constant. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
𝐵 ∈ V    ⇒   ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵
 
18-Sep-2024fvconst0ci 46074 A constant function's value is either the constant or the empty set. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
𝐵 ∈ V    &   𝑌 = ((𝐴 × {𝐵})‘𝑋)    ⇒   (𝑌 = ∅ ∨ 𝑌 = 𝐵)
 
18-Sep-2024fvconstr2 46073 Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   (𝜑 → 𝑋 ∈ (𝐴𝐹𝐵))    ⇒   (𝜑 → 𝐴𝑅𝐵)
 
18-Sep-2024fvconstr 46071 Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   (𝜑 → 𝑌 ∈ 𝑉)    &   (𝜑 → 𝑌 ≠ ∅)    ⇒   (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌))
 
18-Sep-2024f1cof1blem 44455 Lemma for f1cof1b 44456 and focofob 44459. (Contributed by AV, 18-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    &   (𝜑 → ran 𝐹 = 𝐶)    ⇒   (𝜑 → ((𝑃 = 𝐴 ∧ 𝐸 = 𝐶) ∧ (𝑋 = 𝐹 ∧ 𝑌 = 𝐺)))
 
18-Sep-2024fcoresf1lem 44449 Lemma for fcoresf1 44450. (Contributed by AV, 18-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    ⇒   ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = (𝑌‘(𝑋‘𝑍)))
 
18-Sep-2024sbn1ALT 34969 Alternate proof of sbn1 2107, not using the false constant. (Contributed by BJ, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)
 
18-Sep-2024ssltdisj 33942 If 𝐴 preceeds 𝐵, then 𝐴 and 𝐵 are disjoint. (Contributed by Scott Fenton, 18-Sep-2024.)
(𝐴 <<s 𝐵 → (𝐴 ∩ 𝐵) = ∅)
 
18-Sep-2024catcone0 17313 Composition of non-empty hom-sets is non-empty. (Contributed by Zhi Wang, 18-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → (𝑋𝐻𝑌) ≠ ∅)    &   (𝜑 → (𝑌𝐻𝑍) ≠ ∅)    ⇒   (𝜑 → (𝑋𝐻𝑍) ≠ ∅)
 
18-Sep-2024f1cof1 6665 Composition of two one-to-one functions. Generalization of f1co 6666. (Contributed by AV, 18-Sep-2024.)
((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)–1-1→𝐷)
 
18-Sep-2024fcof 6607 Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 6608. (Contributed by AV, 18-Sep-2024.)
((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
 
18-Sep-2024cnvimassrndm 6044 The preimage of a superset of the range of a class is the domain of the class. Generalization of cnvimarndm 5979 for subsets. (Contributed by AV, 18-Sep-2024.)
(ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)
 
18-Sep-2024abeq2w 2816 Version of abeq2 2871 using implicit substitution, which requires fewer axioms. (Contributed by GG and AV, 18-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝜓))
 
17-Sep-2024indthincALT 46222 An alternate proof for indthinc 46221 assuming more axioms including ax-pow 5283 and ax-un 7566. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))    &   (𝜑 → ∅ = (comp‘𝐶))    &   (𝜑 → 𝐶 ∈ 𝑉)    ⇒   (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))
 
17-Sep-20240thinc 46220 The empty category (see 0cat 17315) is thin. (Contributed by Zhi Wang, 17-Sep-2024.)
∅ ∈ ThinCat
 
17-Sep-20240thincg 46219 Any structure with an empty set of objects is a thin category. (Contributed by Zhi Wang, 17-Sep-2024.)
((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat)
 
17-Sep-2024isthincd2 46207 The predicate "𝐶 is a thin category" without knowing 𝐶 is a category (deduction form). The identity arrow operator is also provided as a byproduct. (Contributed by Zhi Wang, 17-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    &   (𝜑 → · = (comp‘𝐶))    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))))    &   ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1 ∈ (𝑦𝐻𝑦))    &   ((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    ⇒   (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 )))
 
17-Sep-2024isthincd 46206 The predicate "is a thin category" (deduction form). (Contributed by Zhi Wang, 17-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    &   (𝜑 → 𝐶 ∈ Cat)    ⇒   (𝜑 → 𝐶 ∈ ThinCat)
 
17-Sep-2024isthincd2lem2 46205 Lemma for isthincd2 46207. (Contributed by Zhi Wang, 17-Sep-2024.)
(𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    &   (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    ⇒   (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐻𝑍))
 
17-Sep-2024thincmo2 46197 Morphisms in the same hom-set are identical. (Contributed by Zhi Wang, 17-Sep-2024.)
(𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐶 ∈ ThinCat)    ⇒   (𝜑 → 𝐹 = 𝐺)
 
17-Sep-2024isthincd2lem1 46196 Lemma for isthincd2 46207 and thincmo2 46197. (Contributed by Zhi Wang, 17-Sep-2024.)
(𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    &   (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    ⇒   (𝜑 → 𝐹 = 𝐺)
 
17-Sep-2024thincssc 46195 A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
ThinCat ⊆ Cat
 
17-Sep-2024thincc 46193 A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
(𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
 
17-Sep-2024isthinc3 46192 A thin category is a category in which, given a pair of objects 𝑥 and 𝑦 and any two morphisms 𝑓, 𝑔 from 𝑥 to 𝑦, the morphisms are equal. (Contributed by Zhi Wang, 17-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔))
 
17-Sep-2024isthinc2 46191 A thin category is a category in which all hom-sets have cardinality less than or equal to the cardinality of 1o. (Contributed by Zhi Wang, 17-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o))
 
17-Sep-2024isthinc 46190 The predicate "is a thin category". (Contributed by Zhi Wang, 17-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
 
17-Sep-2024df-thinc 46189 Definition of the class of thin categories, or posetal categories, whose hom-sets each contain at most one morphism. Example 3.26(2) of [Adamek] p. 33. "ThinCat" was taken instead of "PosCat" because the latter might mean the category of posets. (Contributed by Zhi Wang, 17-Sep-2024.)
ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
 
17-Sep-2024fcoresfo 44452 If a composition is surjective, then the restriction of its first component to the minimum domain is surjective. (Contributed by AV, 17-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    &   (𝜑 → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)    ⇒   (𝜑 → 𝑌:𝐸–onto→𝐷)
 
17-Sep-2024fcores 44448 Every composite function (𝐺 ∘ 𝐹) can be written as composition of restrictions of the composed functions (to their minimum domains). (Contributed by GL and AV, 17-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    ⇒   (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
 
17-Sep-2024fcoreslem4 44447 Lemma 4 for fcores 44448. (Contributed by AV, 17-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    ⇒   (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃)
 
17-Sep-2024fcoreslem2 44445 Lemma 2 for fcores 44448. (Contributed by AV, 17-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    ⇒   (𝜑 → ran 𝑋 = 𝐸)
 
17-Sep-2024fcoreslem1 44444 Lemma 1 for fcores 44448. (Contributed by AV, 17-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    ⇒   (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))
 
17-Sep-2024sltlpss 34014 If two surreals share a birthday, then 𝑋 <s 𝑌 iff the left set of 𝑋 is a proper subset of the left set of 𝑌. (Contributed by Scott Fenton, 17-Sep-2024.)
((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 ↔ ( L ‘𝑋) ⊊ ( L ‘𝑌)))
 
17-Sep-2024lruneq 34013 If two surreals share a birthday, then the union of their left and right sets are equal. (Contributed by Scott Fenton, 17-Sep-2024.)
((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
 
17-Sep-2024cnvimainrn 6926 The preimage of the intersection of the range of a class and a class 𝐴 is the preimage of the class 𝐴. (Contributed by AV, 17-Sep-2024.)
(Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴))
 
17-Sep-2024fncofn 6532 Composition of a function with domain and a function as a function with domain. Generalization of fnco 6533. (Contributed by AV, 17-Sep-2024.)
((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
 
16-Sep-2024neircl 46086 Reverse closure of the neighborhood operation. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by Zhi Wang, 16-Sep-2024.)
(𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝐽 ∈ Top)
 
16-Sep-2024elfvne0 46064 If a function value has a member, then the function is not an empty set (An artifact of our function value definition.) (Contributed by Zhi Wang, 16-Sep-2024.)
(𝐴 ∈ (𝐹‘𝐵) → 𝐹 ≠ ∅)
 
16-Sep-2024isdomn5 40099 The right conjunct in the right hand side of the equivalence of isdomn 20478 is logically equivalent to a less symmetric version where one of the variables is restricted to be nonzero. (Contributed by SN, 16-Sep-2024.)
(∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 ))
 
15-Sep-2024dvdsexpb 40263 dvdssq 16200 generalized to positive integer exponents. (Contributed by SN, 15-Sep-2024.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 ∥ 𝐵 ↔ (𝐴↑𝑁) ∥ (𝐵↑𝑁)))
 
15-Sep-2024dvdsexpnn0 40262 dvdsexpnn 40261 generalized to include zero bases. (Contributed by SN, 15-Sep-2024.)
((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝐴 ∥ 𝐵 ↔ (𝐴↑𝑁) ∥ (𝐵↑𝑁)))
 
15-Sep-2024absdvdsabsb 40248 Divisibility is invariant under taking the absolute value on both sides. (Contributed by SN, 15-Sep-2024.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ (abs‘𝑁)))
 
15-Sep-20240dvds0 40247 0 divides 0. (Contributed by SN, 15-Sep-2024.)
0 ∥ 0
 
15-Sep-2024syl3an12 40103 A double syllogism inference. (Contributed by SN, 15-Sep-2024.)
(𝜑 → 𝜓)    &   (𝜒 → 𝜃)    &   ((𝜓 ∧ 𝜃 ∧ 𝜏) → 𝜂)    ⇒   ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)
 
15-Sep-2024isdomn4 40100 A ring is a domain iff it is nonzero and the cancellation law for multiplication holds. (Contributed by SN, 15-Sep-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎 · 𝑏) = (𝑎 · 𝑐) → 𝑏 = 𝑐)))
 
15-Sep-2024cat1 17728 The definition of category df-cat 17294 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 17725 and setc2ohom 17726 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 17657 and its subsection. (Contributed by Zhi Wang, 15-Sep-2024.)
∃𝑐 ∈ Cat [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
 
15-Sep-2024cat1lem 17727 The category of sets in a "universe" containing the empty set and another set does not have pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. Lemma for cat1 17728. (Contributed by Zhi Wang, 15-Sep-2024.)
𝐶 = (SetCat‘𝑈)    &   (𝜑 → 𝑈 ∈ 𝑉)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → ∅ ∈ 𝑈)    &   (𝜑 → 𝑌 ∈ 𝑈)    &   (𝜑 → ∅ ≠ 𝑌)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
 
15-Sep-2024gcdabs 16166 The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.) (Proof shortened by SN, 15-Sep-2024.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))
 
15-Sep-2024zexpcld 13736 Closure of exponentiation of integers, deductive form. (Contributed by SN, 15-Sep-2024.)
(𝜑 → 𝐴 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → (𝐴↑𝑁) ∈ ℤ)
 
15-Sep-2024fsetexb 8610 The class of all functions from a class 𝐴 into a class 𝐵 is a set iff 𝐵 is a set or 𝐴 is not a set or 𝐴 is empty. (Contributed by AV, 15-Sep-2024.)
({𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V ↔ (𝐴 ∉ V ∨ 𝐴 = ∅ ∨ 𝐵 ∈ V))
 
15-Sep-2024fsetcdmex 8609 The class of all functions from a nonempty set 𝐴 into a class 𝐵 is a set iff 𝐵 is a set . (Contributed by AV, 15-Sep-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∈ V ↔ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V))
 
15-Sep-2024fsetprcnex 8608 The class of all functions from a nonempty set 𝐴 into a proper class 𝐵 is not a set. If one of the preconditions is not fufilled, then {𝑓 ∣ 𝑓:𝐴⟶𝐵} is a set, see fsetdmprc0 8601 for 𝐴 ∉ V, fset0 8600 for 𝐴 = ∅, and fsetex 8602 for 𝐵 ∈ V, see also fsetexb 8610. (Contributed by AV, 14-Sep-2024.) (Proof shortened by BJ, 15-Sep-2024.)
(((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)
 
15-Sep-2024fsetfocdm 8607 The class of functions with a given domain that is a set and a given codomain is mapped, through evaluation at a point of the domain, onto the codomain. (Contributed by AV, 15-Sep-2024.)
𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵}    &   𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑆:𝐹–onto→𝐵)
 
15-Sep-2024fsetfcdm 8606 The class of functions with a given domain and a given codomain is mapped, through evaluation at a point of the domain, into the codomain. (Contributed by AV, 15-Sep-2024.)
𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵}    &   𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋))    ⇒   (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵)
 
14-Sep-2024fsetprcnexALT 44443 First version of proof for fsetprcnex 8608, which was much more complicated. (Contributed by AV, 14-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)
 
14-Sep-2024cfsetsnfsetf1o 44442 The mapping of the class of singleton functions into the class of constant functions is a bijection. (Contributed by AV, 14-Sep-2024.)
𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1-onto→𝐹)
 
14-Sep-2024cfsetsnfsetfo 44441 The mapping of the class of singleton functions into the class of constant functions is a surjection. (Contributed by AV, 14-Sep-2024.)
𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–onto→𝐹)
 
14-Sep-2024cfsetsnfsetf1 44440 The mapping of the class of singleton functions into the class of constant functions is an injection. (Contributed by AV, 14-Sep-2024.)
𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1→𝐹)
 
14-Sep-2024cfsetsnfsetf 44439 The mapping of the class of singleton functions into the class of constant functions is a function. (Contributed by AV, 14-Sep-2024.)
𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)
 
13-Sep-2024fcoreslem3 44446 Lemma 3 for fcores 44448. (Contributed by AV, 13-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    ⇒   (𝜑 → 𝑋:𝑃–onto→𝐸)
 
13-Sep-2024cfsetsnfsetfv 44438 The function value of the mapping of the class of singleton functions into the class of constant functions. (Contributed by AV, 13-Sep-2024.)
𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝐻‘𝑋) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))
 
13-Sep-2024cfsetssfset 44437 The class of constant functions is a subclass of the class of functions. (Contributed by AV, 13-Sep-2024.)
𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    ⇒   𝐹 ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}
 
13-Sep-2024fsetsnprcnex 44436 The class of all functions from a (proper) singleton into a proper class 𝐵 is not a set. (Contributed by AV, 13-Sep-2024.)
((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ∉ V)
 
13-Sep-2024fsetsnf1o 44435 The mapping of an element of a class to a singleton function is a bijection. (Contributed by AV, 13-Sep-2024.)
𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1-onto→𝐴)
 
13-Sep-2024fsetsnfo 44434 The mapping of an element of a class to a singleton function is a surjection. (Contributed by AV, 13-Sep-2024.)
𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   (𝑆 ∈ 𝑉 → 𝐹:𝐵–onto→𝐴)
 
13-Sep-2024fsetsnf1 44433 The mapping of an element of a class to a singleton function is an injection. (Contributed by AV, 13-Sep-2024.)
𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1→𝐴)
 
13-Sep-2024fsetsnf 44432 The mapping of an element of a class to a singleton function is a function. (Contributed by AV, 13-Sep-2024.)
𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)
 
13-Sep-2024fsetabsnop 44431 The class of all functions from a (proper) singleton into 𝐵 is the class of all the singletons of (proper) ordered pairs over the elements of 𝐵 as second component. (Contributed by AV, 13-Sep-2024.)
(𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
 
13-Sep-2024fsetsniunop 44430 The class of all functions from a (proper) singleton into 𝐵 is the union of all the singletons of (proper) ordered pairs over the elements of 𝐵 as second component. (Contributed by AV, 13-Sep-2024.)
(𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}})
 
13-Sep-2024fset0 8600 The set of functions from the empty set is the singleton containing the empty set. (Contributed by AV, 13-Sep-2024.)
{𝑓 ∣ 𝑓:∅⟶𝐵} = {∅}
 
13-Sep-2024fsetsspwxp 8599 The class of all functions from 𝐴 into 𝐵 is a subclass of the power class of the cartesion product of 𝐴 and 𝐵. (Contributed by AV, 13-Sep-2024.)
{𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
 
12-Sep-2024fineqvpow 32965 If the Axiom of Infinity is negated, then the Axiom of Power Sets becomes redundant. (Contributed by BTernaryTau, 12-Sep-2024.)
(Fin = V → ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
 
12-Sep-2024fineqvrep 32964 If the Axiom of Infinity is negated, then the Axiom of Replacement becomes redundant. (Contributed by BTernaryTau, 12-Sep-2024.)
(Fin = V → (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
 
10-Sep-2024iscnrm3v 46135 A topology is completely normal iff two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 10-Sep-2024.)
(𝐽 ∈ Top → (𝐽 ∈ CNrm ↔ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
 
10-Sep-2024onunel 33592 The union of two ordinals is in a third iff both of the first two are. (Contributed by Scott Fenton, 10-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
 
10-Sep-2024entrfil 8931 Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8747). (Contributed by BTernaryTau, 10-Sep-2024.)
((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
 
9-Sep-2024seppcld 46111 If two sets are precisely separated by a continuous function, then they are closed. An alternate proof involves II ∈ Fre. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})))    ⇒   (𝜑 → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽)))
 
9-Sep-2024seppsepf 46110 If two sets are precisely separated by a continuous function, then they are separated by the continuous function. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})))    ⇒   (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))
 
9-Sep-2024sepfsepc 46109 If two sets are separated by a continuous function, then they are separated by closed neighborhoods. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))    ⇒   (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))
 
9-Sep-2024io1ii 46102 (𝐴(,]1) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
(0 ≤ 𝐴 → (𝐴(,]1) ∈ II)
 
9-Sep-2024i0oii 46101 (0[,)𝐴) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝐴 ≤ 1 → (0[,)𝐴) ∈ II)
 
9-Sep-2024iooii 46099 Open intervals are open sets of II. (Contributed by Zhi Wang, 9-Sep-2024.)
((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴(,)𝐵) ∈ II)
 
9-Sep-2024cnneiima 46098 Given a continuous function, the preimage of a neighborhood is a neighborhood. To be precise, the preimage of a neighborhood of a subset 𝑇 of the codomain of a continuous function is a neighborhood of any subset of the preimage of 𝑇. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑 → 𝑁 ∈ ((nei‘𝐾)‘𝑇))    &   (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝑇))    ⇒   (𝜑 → (◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆))
 
9-Sep-2024iccdisj 46080 If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
 
9-Sep-2024iccdisj2 46079 If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
((𝐴 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
 
9-Sep-2024iccin 46078 Intersection of two closed intervals of extended reals. (Contributed by Zhi Wang, 9-Sep-2024.)
(((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)[,]if(𝐵 ≤ 𝐷, 𝐵, 𝐷)))
 
9-Sep-2024predisj 46044 Preimages of disjoint sets are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → Fun 𝐹)    &   (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝐴))    &   (𝜑 → 𝑇 ⊆ (◡𝐹 “ 𝐵))    ⇒   (𝜑 → (𝑆 ∩ 𝑇) = ∅)
 
9-Sep-2024naddss2 33769 Ordinal less-than-or-equal is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +no 𝐴) ⊆ (𝐶 +no 𝐵)))
 
9-Sep-2024naddss1 33768 Ordinal less-than-or-equal is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))
 
9-Sep-2024naddel2 33767 Ordinal less-than is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐶 +no 𝐴) ∈ (𝐶 +no 𝐵)))
 
9-Sep-2024naddel1 33766 Ordinal less-than is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶)))
 
9-Sep-2024naddelim 33765 Ordinal less-than is preserved by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶)))
 
9-Sep-2024ensymfib 8930 Symmetry of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike ensymb 8743). (Contributed by BTernaryTau, 9-Sep-2024.)
(𝐴 ∈ Fin → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴))
 
9-Sep-2024f1oenfirn 8927 If the range of a one-to-one, onto function is finite, then the domain and range of the function are equinumerous. (Contributed by BTernaryTau, 9-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
 
9-Sep-2024cnvfi 8924 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5283. (Revised by BTernaryTau, 9-Sep-2024.)
(𝐴 ∈ Fin → ◡𝐴 ∈ Fin)
 
9-Sep-2024f1dom3g 8710 The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 8715 does not require the Axiom of Replacement nor the Axiom of Power Sets. (Contributed by BTernaryTau, 9-Sep-2024.)
((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
 
8-Sep-2024sepcsepo 46108 If two sets are separated by closed neighborhoods, then they are separated by (open) neighborhoods. See sepnsepo 46105 for the equivalence between separatedness by open neighborhoods and separatedness by neighborhoods. Although 𝐽 ∈ Top might be redundant here, it is listed for explicitness. 𝐽 ∈ Top can be obtained from neircl 46086, adantr 480, and rexlimiva 3209. (Contributed by Zhi Wang, 8-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))    ⇒   (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
 
8-Sep-2024icccldii 46100 Closed intervals are closed sets of II. Note that iccss 13076, iccordt 22273, and ordtresticc 22282 are proved from ixxss12 13028, ordtcld3 22258, and ordtrest2 22263, respectively. An alternate proof uses restcldi 22232, dfii2 23951, and icccld 23836. (Contributed by Zhi Wang, 8-Sep-2024.)
((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴[,]𝐵) ∈ (Clsd‘II))
 
8-Sep-2024enreffi 8929 Equinumerosity is reflexive for finite sets, proved without using the Axiom of Power Sets (unlike enrefg 8727). (Contributed by BTernaryTau, 8-Sep-2024.)
(𝐴 ∈ Fin → 𝐴 ≈ 𝐴)
 
8-Sep-2024f1oenfi 8926 If the domain of a one-to-one, onto function is finite, then the domain and range of the function are equinumerous. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1oeng 8714). (Contributed by BTernaryTau, 8-Sep-2024.)
((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
 
8-Sep-2024relopabv 5720 A class of ordered pairs is a relation. For a version without a disjoint variable condition, but using ax-11 2156 and ax-12 2173, see relopab 5723. (Contributed by SN, 8-Sep-2024.)
Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
8-Sep-2024ab0 4305 The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 4311 (from which it could be proved using as many essential proof steps but one fewer syntactic step, at the cost of depending on df-ne 2943). (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2817, ax-8 2110. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by SN, 8-Sep-2024.)
({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
8-Sep-2024ceqsralv 3459 Restricted quantifier version of ceqsalv 3457. (Contributed by NM, 21-Jun-2013.) Avoid ax-9 2118, ax-12 2173, ax-ext 2709. (Revised by SN, 8-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
 
8-Sep-2024ceqsalv 3457 A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) Avoid ax-12 2173. (Revised by SN, 8-Sep-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
 
7-Sep-2024seposep 46107 If two sets are separated by (open) neighborhoods, then they are separated subsets of the underlying set. Note that separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. See sepnsepo 46105. The relationship between separatedness and closure is also seen in isnrm 22394, isnrm2 22417, isnrm3 22418. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))    ⇒   (𝜑 → ((𝑆 ⊆ ∪ 𝐽 ∧ 𝑇 ⊆ ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))
 
7-Sep-2024sepdisj 46106 Separated sets are disjoint. Note that in general separatedness also requires 𝑇 ⊆ ∪ 𝐽 and (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ as well but they are unnecessary here. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)    ⇒   (𝜑 → (𝑆 ∩ 𝑇) = ∅)
 
7-Sep-2024ssdisjdr 46042 Subset preserves disjointness. Deduction form of ssdisj 4390. Alternatively this could be proved with ineqcom 4133 in tandem with ssdisjd 46041. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → (𝐶 ∩ 𝐵) = ∅)    ⇒   (𝜑 → (𝐶 ∩ 𝐴) = ∅)
 
7-Sep-2024ssdisjd 46041 Subset preserves disjointness. Deduction form of ssdisj 4390. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → (𝐵 ∩ 𝐶) = ∅)    ⇒   (𝜑 → (𝐴 ∩ 𝐶) = ∅)
 
7-Sep-2024naddssim 33764 Ordinal less-than-or-equal is preserved by natural addition. (Contributed by Scott Fenton, 7-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))
 
7-Sep-2024pwfi 8923 The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5283. (Revised by BTernaryTau, 7-Sep-2024.)
(𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
 
7-Sep-2024pwfilem 8922 Lemma for pwfi 8923. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5283. (Revised by BTernaryTau, 7-Sep-2024.)
𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))    ⇒   (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
 
7-Sep-2024pwfir 8921 If the power set of a set is finite, then the set itself is finite. (Contributed by BTernaryTau, 7-Sep-2024.)
(𝒫 𝐵 ∈ Fin → 𝐵 ∈ Fin)
 
7-Sep-2024imafi 8920 Images of finite sets are finite. For a shorter proof using ax-pow 5283, see imafiALT 9042. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5283. (Revised by BTernaryTau, 7-Sep-2024.)
((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin)
 
6-Sep-2024clddisj 46085 Two ways of saying that two closed sets are disjoint, if 𝐽 is a topology and 𝑋 is a closed set. An alternative proof is similar to that of opndisj 46084 with elssuni 4868 replaced by the combination of cldss 22088 and eqid 2738. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
 
6-Sep-2024opndisj 46084 Two ways of saying that two open sets are disjoint, if 𝐽 is a topology and 𝑋 is an open set. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ (𝐽 ∩ 𝒫 𝑍) ↔ (𝑌 ∈ 𝐽 ∧ (𝑋 ∩ 𝑌) = ∅)))
 
6-Sep-2024clduni 46082 The union of closed sets is the underlying set of the topology (the union of open sets). (Contributed by Zhi Wang, 6-Sep-2024.)
(𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽)
 
6-Sep-2024r19.41dv 46035 A complex deduction form of r19.41v 3273. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))
 
6-Sep-2024ralbidb 46033 Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc 46034 for a more generalized form. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)))    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 ↔ 𝜃))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃)))
 
6-Sep-2024pm5.32dra 46028 Reverse distribution of implication over biconditional (deduction form). (Contributed by Zhi Wang, 6-Sep-2024.)
(𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))    ⇒   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
 
6-Sep-2024eq0rdv 4335 Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.) Avoid ax-8 2110, df-clel 2817. (Revised by Gino Giotto, 6-Sep-2024.)
(𝜑 → ¬ 𝑥 ∈ 𝐴)    ⇒   (𝜑 → 𝐴 = ∅)
 
6-Sep-2024eq0 4274 A class is equal to the empty set if and only if it has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2156, ax-12 2173. (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2110, df-clel 2817. (Revised by Gino Giotto, 6-Sep-2024.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
 
6-Sep-2024vn0 4269 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2110, df-clel 2817. (Revised by Gino Giotto, 6-Sep-2024.)
V ≠ ∅
 
5-Sep-2024iscnrm4 46136 A completely normal topology is a topology in which two separated sets can be separated by neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛 ∩ 𝑚) = ∅)))
 
5-Sep-2024iscnrm3 46134 A completely normal topology is a topology in which two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
 
5-Sep-2024iscnrm3l 46133 Lemma for iscnrm3 46134. Given a topology 𝐽, if two separated sets can be separated by open neighborhoods, then all subspaces of the topology 𝐽 are normal, i.e., two disjoint closed sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝐽 ∈ Top → (∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) → ((𝐶 ∩ 𝐷) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))))
 
5-Sep-2024iscnrm3llem2 46132 Lemma for iscnrm3l 46133. If there exist disjoint open neighborhoods in the original topology for two disjoint closed sets in a subspace, then they can be separated by open neighborhoods in the subspace topology. (Could shorten proof with ssin0 42492.) (Contributed by Zhi Wang, 5-Sep-2024.)
((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))
 
5-Sep-2024iscnrm3r 46130 Lemma for iscnrm3 46134. If all subspaces of a topology are normal, i.e., two disjoint closed sets can be separated by open neighborhoods, then in the original topology two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 ∪ 𝐽∀𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → ((𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) → (((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))))
 
5-Sep-2024iscnrm3rlem8 46129 Lemma for iscnrm3r 46130. Disjoint open neighborhoods in the subspace topology are disjoint open neighborhoods in the original topology given that the subspace is an open set in the original topology. Therefore, given any two sets separated in the original topology and separated by open neighborhoods in the subspace topology, they must be separated by open neighborhoods in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.)
((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) → (∃𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
 
5-Sep-2024iscnrm3rlem7 46128 Lemma for iscnrm3rlem8 46129. Open neighborhoods in the subspace topology are open neighborhoods in the original topology given that the subspace is an open set in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   (𝜑 → 𝑇 ⊆ ∪ 𝐽)    &   (𝜑 → 𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))    ⇒   (𝜑 → 𝑂 ∈ 𝐽)
 
5-Sep-2024iscnrm3rlem6 46127 Lemma for iscnrm3rlem7 46128. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   (𝜑 → 𝑇 ⊆ ∪ 𝐽)    &   (𝜑 → 𝑂 ⊆ (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))    ⇒   (𝜑 → (𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ↔ 𝑂 ∈ 𝐽))
 
5-Sep-2024iscnrm3rlem5 46126 Lemma for iscnrm3rlem6 46127. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   (𝜑 → 𝑇 ⊆ ∪ 𝐽)    ⇒   (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽)
 
5-Sep-2024iscnrm3rlem4 46125 Lemma for iscnrm3rlem8 46129. Given two disjoint subsets 𝑆 and 𝑇 of the underlying set of a topology 𝐽, if 𝑁 is a superset of (((cls‘𝐽)‘𝑆) ∖ 𝑇), then it is a superset of 𝑆. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ⊆ 𝑁)    ⇒   (𝜑 → 𝑆 ⊆ 𝑁)
 
5-Sep-2024iscnrm3rlem3 46124 Lemma for iscnrm3r 46130. The designed subspace is a subset of the original set; the two sets are closed sets in the subspace. (Contributed by Zhi Wang, 5-Sep-2024.)
((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → ((∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 ∪ 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))))
 
5-Sep-2024iscnrm3rlem2 46123 Lemma for iscnrm3rlem3 46124. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ ∪ 𝐽)    ⇒   (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ 𝑇)))))
 
5-Sep-2024iscnrm3rlem1 46122 Lemma for iscnrm3rlem2 46123. The hypothesis could be generalized to (𝜑 → (𝑆 ∖ 𝑇) ⊆ 𝑋). (Contributed by Zhi Wang, 5-Sep-2024.)
(𝜑 → 𝑆 ⊆ 𝑋)    ⇒   (𝜑 → (𝑆 ∖ 𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))))
 
5-Sep-2024iscnrm3lem7 46121 Lemma for iscnrm3rlem8 46129 and iscnrm3llem2 46132 involving restricted existential quantifications. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝑧 = 𝑍 → (𝜒 ↔ 𝜃))    &   (𝑤 = 𝑊 → (𝜃 ↔ 𝜏))    &   ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → (𝑍 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝜏))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒))
 
5-Sep-2024iscnrm3lem6 46120 Lemma for iscnrm3lem7 46121. (Contributed by Zhi Wang, 5-Sep-2024.)
((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) ∧ 𝜓) → 𝜒)    ⇒   (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑊 𝜓 → 𝜒))
 
5-Sep-2024disjdifb 46043 Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.)
((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅
 
4-Sep-2024iscnrm3llem1 46131 Lemma for iscnrm3l 46133. Closed sets in the subspace are subsets of the underlying set of the original topology. (Contributed by Zhi Wang, 4-Sep-2024.)
((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (𝐶 ∈ 𝒫 ∪ 𝐽 ∧ 𝐷 ∈ 𝒫 ∪ 𝐽))
 
4-Sep-2024iscnrm3lem4 46118 Lemma for iscnrm3lem5 46119 and iscnrm3r 46130. (Contributed by Zhi Wang, 4-Sep-2024.)
(𝜂 → (𝜓 → 𝜁))    &   ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜂)    &   ((𝜑 ∧ 𝜒 ∧ 𝜃) → (𝜁 → 𝜏))    ⇒   (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
 
4-Sep-2024iscnrm3lem3 46117 Lemma for iscnrm3lem4 46118. (Contributed by Zhi Wang, 4-Sep-2024.)
(((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓))
 
4-Sep-2024on3ind 33756 Triple induction over ordinals. (Contributed by Scott Fenton, 4-Sep-2024.)
(𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜃 ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 𝜒 ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 𝜁) ∧ (∀𝑑 ∈ 𝑎 𝜓 ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜏 ∧ ∀𝑒 ∈ 𝑏 𝜎) ∧ ∀𝑓 ∈ 𝑐 𝜂) → 𝜑))    ⇒   ((𝑋 ∈ On ∧ 𝑌 ∈ On ∧ 𝑍 ∈ On) → 𝜆)
 
4-Sep-2024xpord3ind 33727 Induction over the triple cross product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 4-Sep-2024.)
𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   𝑅 Fr 𝐴    &   𝑅 Po 𝐴    &   𝑅 Se 𝐴    &   𝑆 Fr 𝐵    &   𝑆 Po 𝐵    &   𝑆 Se 𝐵    &   𝑇 Fr 𝐶    &   𝑇 Po 𝐶    &   𝑇 Se 𝐶    &   (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑))    ⇒   ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝜆)
 
4-Sep-2024vex 3426 All setvar variables are sets (see isset 3435). Theorem 6.8 of [Quine] p. 43. A shorter proof is possible from eleq2i 2830 but it uses more axioms. (Contributed by NM, 26-May-1993.) Remove use of ax-12 2173. (Revised by SN, 28-Aug-2023.) (Proof shortened by BJ, 4-Sep-2024.)
𝑥 ∈ V
 
3-Sep-2024iscnrm3lem5 46119 Lemma for iscnrm3l 46133. (Contributed by Zhi Wang, 3-Sep-2024.)
((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → (𝜑 ↔ 𝜓))    &   ((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → (𝜒 ↔ 𝜃))    &   ((𝜏 ∧ 𝜂 ∧ 𝜁) → (𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊))    &   ((𝜏 ∧ 𝜂 ∧ 𝜁) → ((𝜓 → 𝜃) → 𝜎))    ⇒   (𝜏 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 (𝜑 → 𝜒) → (𝜂 → (𝜁 → 𝜎))))
 
3-Sep-2024iscnrm3lem2 46116 Lemma for iscnrm3 46134 proving a biconditional on restricted universal quantifications. (Contributed by Zhi Wang, 3-Sep-2024.)
(𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 → ((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒)))    &   (𝜑 → (∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓)))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 ↔ ∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒))
 
3-Sep-2024iscnrm3lem1 46115 Lemma for iscnrm3 46134. Subspace topology is a topology. (Contributed by Zhi Wang, 3-Sep-2024.)
(𝐽 ∈ Top → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ((𝐽 ↾t 𝑥) ∈ Top ∧ 𝜑)))
 
3-Sep-2024exp12bd 46029 The import-export theorem (impexp 450) for biconditionals (deduction form). (Contributed by Zhi Wang, 3-Sep-2024.)
(𝜑 → (((𝜓 ∧ 𝜒) → 𝜃) ↔ ((𝜏 ∧ 𝜂) → 𝜁)))    ⇒   (𝜑 → ((𝜓 → (𝜒 → 𝜃)) ↔ (𝜏 → (𝜂 → 𝜁))))
 
3-Sep-2024on2recsov 33754 Calculate the value of the double ordinal recursion operator. (Contributed by Scott Fenton, 3-Sep-2024.)
𝐹 = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), 𝐺)    ⇒   ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ ((suc 𝐴 × suc 𝐵) ∖ {⟨𝐴, 𝐵⟩}))))
 
3-Sep-2024on2recsfn 33753 Show that double recursion over ordinals yields a function over pairs of ordinals. (Contributed by Scott Fenton, 3-Sep-2024.)
𝐹 = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), 𝐺)    ⇒   𝐹 Fn (On × On)
 
2-Sep-2024dfnrm3 46114 A topological space is normal if any disjoint closed sets can be separated by neighborhoods. An alternate definition of df-nrm 22376. (Contributed by Zhi Wang, 2-Sep-2024.)
Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝑗)‘𝑐)∃𝑦 ∈ ((nei‘𝑗)‘𝑑)(𝑥 ∩ 𝑦) = ∅)}
 
2-Sep-2024restclssep 46097 Two disjoint closed sets in a subspace topology are separated in the original topology. (Contributed by Zhi Wang, 2-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑋 = ∪ 𝐽)    &   (𝜑 → 𝑌 ⊆ 𝑋)    &   (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    &   (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   (𝜑 → 𝑇 ∈ (Clsd‘𝐾))    ⇒   (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))
 
2-Sep-2024restclsseplem 46096 Lemma for restclssep 46097. (Contributed by Zhi Wang, 2-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑋 = ∪ 𝐽)    &   (𝜑 → 𝑌 ⊆ 𝑋)    &   (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    &   (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   (𝜑 → 𝑇 ⊆ 𝑌)    ⇒   (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
 
2-Sep-2024restcls2 46095 A closed set in a subspace topology is the closure in the original topology intersecting with the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑋 = ∪ 𝐽)    &   (𝜑 → 𝑌 ⊆ 𝑋)    &   (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    ⇒   (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
 
2-Sep-2024restcls2lem 46094 A closed set in a subspace topology is a subset of the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑋 = ∪ 𝐽)    &   (𝜑 → 𝑌 ⊆ 𝑋)    &   (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    ⇒   (𝜑 → 𝑆 ⊆ 𝑌)
 
2-Sep-2024elrab2w 40095 Membership in a restricted class abstraction. This is to elrab2 3620 what elab2gw 40094 is to elab2g 3604. (Contributed by SN, 2-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))    &   𝐶 = {𝑥 ∈ 𝐵 ∣ 𝜑}    ⇒   (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜒))
 
2-Sep-2024ralf0 4441 The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) Avoid df-clel 2817, ax-8 2110. (Revised by Gino Giotto, 2-Sep-2024.)
¬ 𝜑    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)
 
2-Sep-2024ral0 4440 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) Avoid df-clel 2817, ax-8 2110. (Revised by Gino Giotto, 2-Sep-2024.)
∀𝑥 ∈ ∅ 𝜑
 
2-Sep-2024ralidm 4439 Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) Reduce axiom usage. (Revised by Gino Giotto, 2-Sep-2024.)
(∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
 
2-Sep-2024rexn0 4438 Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2817, ax-8 2110. (Revised by Gino Giotto, 2-Sep-2024.)
(∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
 
2-Sep-2024rzal 4436 Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid df-clel 2817, ax-8 2110. (Revised by Gino Giotto, 2-Sep-2024.)
(𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
 
2-Sep-2024sbc5 3739 An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by SN, 2-Sep-2024.)
([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
 
2-Sep-2024vtocld 3484 Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-10 2139, ax-11 2156, ax-12 2173. (Revised by SN, 2-Sep-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    &   (𝜑 → 𝜓)    ⇒   (𝜑 → 𝜒)
 
2-Sep-2024clelab 2882 Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) Avoid ax-11 2156, see sbc5ALT 3740 for more details. (Revised by SN, 2-Sep-2024.)
(𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
 
1-Sep-2024isnrm4 46112 A topological space is normal iff any two disjoint closed sets are separated by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
(𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥 ∩ 𝑦) = ∅)))
 
1-Sep-2024sepnsepo 46105 Open neighborhood and neighborhood is equivalent regarding disjointness for both sides. Namely, separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    ⇒   (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
 
1-Sep-2024sepnsepolem2 46104 Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 46105. Proof could be shortened by 1 step using ssdisjdr 46042. (Contributed by Zhi Wang, 1-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    ⇒   (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
 
1-Sep-2024sepnsepolem1 46103 Lemma for sepnsepo 46105. (Contributed by Zhi Wang, 1-Sep-2024.)
(∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥 ∈ 𝐽 (𝜑 ∧ ∃𝑦 ∈ 𝐽 (𝜓 ∧ 𝜒)))
 
1-Sep-2024ruvALT 40096 Alternate proof of ruv 9291 with one fewer syntax step thanks to using elirrv 9285 instead of elirr 9286. However, it does not change the compressed proof size or the number of symbols in the generated display, so it is not considered a shortening according to conventions 28665. (Contributed by SN, 1-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
{𝑥 ∣ 𝑥 ∉ 𝑥} = V
 
1-Sep-2024bj-clel3gALT 35148 Alternate proof of clel3g 3584. (Contributed by BJ, 1-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)))
 
1-Sep-2024vopelopabsb 5435 The law of concretion in terms of substitutions. Version of opelopabsb 5436 with set variables. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Remove unnecessary commutation. (Revised by SN, 1-Sep-2024.)
(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
 
1-Sep-2024copsex2g 5401 Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) Use a similar proof to copsex4g 5403 to reduce axiom usage. (Revised by SN, 1-Sep-2024.)
((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
 
1-Sep-2024intpr 4910 The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.) Prove from intprg 4909. (Revised by BJ, 1-Sep-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    ⇒   ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)
 
1-Sep-2024intprg 4909 The intersection of a pair is the intersection of its members. Closed form of intpr 4910. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.) (Proof shortened by BJ, 1-Sep-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))
 
1-Sep-2024unipr 4854 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.) (Proof shortened by BJ, 1-Sep-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    ⇒   ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)
 
1-Sep-2024uniprg 4853 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.) Avoid using unipr 4854 to prove it from uniprg 4853. (Revised by BJ, 1-Sep-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
 
1-Sep-2024clel4g 3586 Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent and avoid ax-12 2173. (Revised by BJ, 1-Sep-2024.)
(𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)))
 
1-Sep-2024clel2g 3581 Alternate definition of membership when the member is a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent. (Revised by BJ, 12-Feb-2022.) Avoid ax-12 2173. (Revised by BJ, 1-Sep-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)))
 
31-Aug-2024opnneieqvv 46093 The equivalence between neighborhood and open neighborhood. A variant of opnneieqv 46092 with two dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    &   ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
 
31-Aug-2024opnneieqv 46092 The equivalence between neighborhood and open neighborhood. See opnneieqvv 46093 for different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    &   ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓)))
 
31-Aug-2024opnneil 46091 A variant of opnneilv 46090. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    &   ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓)))
 
31-Aug-2024opnneilv 46090 The converse of opnneir 46088 with different dummy variables. Note that the second hypothesis could be generalized by adding 𝑦 ∈ 𝐽 to the antecedent. See the proof for details. Although 𝐽 ∈ Top might be redundant here (see neircl 46086), it is listed for explicitness. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    &   ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
 
31-Aug-2024opnneirv 46089 A variant of opnneir 46088 with different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) → ∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝜒))
 
31-Aug-2024opnneir 46088 If something is true for an open neighborhood, it must be true for a neighborhood. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    ⇒   (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓))
 
31-Aug-2024opnneilem 46087 Lemma factoring out common proof steps of opnneil 46091 and opnneirv 46089. (Contributed by Zhi Wang, 31-Aug-2024.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
 
31-Aug-2024dfaiota3 44471 Alternate definition of ℩': this is to df-aiota 44464 what dfiota4 6410 is to df-iota 6376. operation using the if operator. It is simpler than df-aiota 44464 and uses no dummy variables, so it would be the preferred definition if ℩' becomes the description binder used in set.mm. (Contributed by BJ, 31-Aug-2024.)
(℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V)
 
31-Aug-2024aiotaint 44470 This is to df-aiota 44464 what iotauni 6393 is to df-iota 6376 (it uses intersection like df-aiota 44464, similar to iotauni 6393 using union like df-iota 6376; we could also prove an analogous result using union here too, in the same way that we have iotaint 6394). (Contributed by BJ, 31-Aug-2024.)
(∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
 
31-Aug-2024acos1half 40098 The arccosine of 1 / 2 is π / 3. (Contributed by SN, 31-Aug-2024.)
(arccos‘(1 / 2)) = (π / 3)
 
31-Aug-2024f1ofvswap 7158 Swapping two values in a bijection between two classes yields another bijection between those classes. (Contributed by BTernaryTau, 31-Aug-2024.)
((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}):𝐴–1-1-onto→𝐵)
 
30-Aug-2024dfnrm2 46113 A topological space is normal if any disjoint closed sets can be separated by open neighborhoods. An alternate definition of df-nrm 22376. (Contributed by Zhi Wang, 30-Aug-2024.)
Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝑗 ∃𝑦 ∈ 𝑗 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))}
 
30-Aug-2024opncldeqv 46083 Conditions on open sets are equivalent to conditions on closed sets. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    &   ((𝜑 ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))
 
30-Aug-2024ralbidc 46034 Formula-building rule for restricted universal quantifier and additional condition (deduction form). A variant of ralbidb 46033. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)))    &   (𝜑 → ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → (𝜒 ↔ 𝜃)))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃)))
 
30-Aug-2024pm5.32dav 46027 Distribution of implication over biconditional (deduction form). Variant of pm5.32da 578. (Contributed by Zhi Wang, 30-Aug-2024.)
((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜓)))
 
30-Aug-2024logic2 46026 Variant of logic1 46024. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    &   (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)))    ⇒   (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
 
30-Aug-2024logic1a 46025 Variant of logic1 46024. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    &   ((𝜑 ∧ 𝜓) → (𝜃 ↔ 𝜏))    ⇒   (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
 
30-Aug-2024logic1 46024 Distribution of implication over biconditional with replacement (deduction form). (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    &   (𝜑 → (𝜓 → (𝜃 ↔ 𝜏)))    ⇒   (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
 
30-Aug-2024pm4.71da 46023 Deduction converting a biconditional to a biconditional with conjunction. Variant of pm4.71d 561. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))
 
30-Aug-2024abn0 4311 Nonempty class abstraction. See also ab0 4305. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) Avoid df-clel 2817, ax-8 2110. (Revised by Gino Giotto, 30-Aug-2024.)
({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
 
30-Aug-2024ab0orv 4309 The class abstraction defined by a formula not containing the abstraction variable is either the empty set or the universal class. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.) Reduce axiom usage. (Revised by Gino Giotto, 30-Aug-2024.)
({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)
 
30-Aug-2024ab0OLD 4306 Obsolete version of ab0 4305 as of 8-Sep-2024. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2817, ax-8 2110. (Revised by Gino Giotto, 30-Aug-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
30-Aug-2024rru 3709 Relative version of Russell's paradox ru 3710 (which corresponds to the case 𝐴 = V).

Originally a subproof in pwnss 5267. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3049. (Revised by Steven Nguyen, 23-Nov-2022.) Reduce axiom usage. (Revised by Gino Giotto, 30-Aug-2024.)

¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴
 
30-Aug-2024abv 3433 The class of sets verifying a property is the universal class if and only if that property is a tautology. The reverse implication (bj-abv 35018) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2817, ax-8 2110. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by BJ, 30-Aug-2024.)
({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)
 
29-Aug-2024dftermo3 17637 An alternate definition of df-termo 17616 depending on df-inito 17615, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.)
TermO = (InitO ∘ (oppCat ↾ Cat))
 
29-Aug-2024dfinito3 17636 An alternate definition of df-inito 17615 depending on df-termo 17616, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.)
InitO = (TermO ∘ (oppCat ↾ Cat))
 
29-Aug-2024dftermo2 17635 A terminal object is an initial object in the opposite category. An alternate definition of df-termo 17616 depending on df-inito 17615. (Contributed by Zhi Wang, 29-Aug-2024.)
TermO = (𝑐 ∈ Cat ↦ (InitO‘(oppCat‘𝑐)))
 
29-Aug-2024dfinito2 17634 An initial object is a terminal object in the opposite category. An alternate definition of df-inito 17615 depending on df-termo 17616. (Contributed by Zhi Wang, 29-Aug-2024.)
InitO = (𝑐 ∈ Cat ↦ (TermO‘(oppCat‘𝑐)))
 
29-Aug-2024zeroofn 17620 ZeroO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
ZeroO Fn Cat
 
29-Aug-2024termofn 17619 TermO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
TermO Fn Cat
 
29-Aug-2024initofn 17618 InitO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
InitO Fn Cat
 
29-Aug-2024oppccatf 17356 oppCat restricted to Cat is a function from Cat to Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
(oppCat ↾ Cat):Cat⟶Cat
 
27-Aug-2024nmfval0 23652 The value of the norm function on a structure containing a zero as the distance restricted to the elements of the base set to zero. Examples of structures containing a "zero" are groups (see nmfval2 23653 proved from this theorem and grpidcl 18522) or more generally monoids (see mndidcl 18315), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 23653. (Revised by BJ, 27-Aug-2024.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &    0 = (0g‘𝑊)    &   𝐷 = (dist‘𝑊)    &   𝐸 = (𝐷 ↾ (𝑋 × 𝑋))    ⇒   ( 0 ∈ 𝑋 → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
 
26-Aug-2024naddid1 33763 Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
(𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
 
26-Aug-2024naddcom 33762 Natural addition commutes. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))
 
26-Aug-2024naddov2 33761 Alternate expression for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})
 
26-Aug-2024naddov 33760 The value of natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
 
26-Aug-2024naddcl 33759 Closure law for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) ∈ On)
 
26-Aug-2024naddcllem 33758 Lemma for ordinal addition closure. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +no 𝐵) ∈ On ∧ (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)}))
 
26-Aug-2024naddfn 33757 Natural addition is a function over pairs of ordinals. (Contributed by Scott Fenton, 26-Aug-2024.)
+no Fn (On × On)
 
26-Aug-2024on2ind 33755 Double induction over ordinal numbers. (Contributed by Scott Fenton, 26-Aug-2024.)
(𝑎 = 𝑐 → (𝜑 ↔ 𝜓))    &   (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))    &   (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))    &   (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))    &   (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))    &   ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒 ∧ ∀𝑐 ∈ 𝑎 𝜓 ∧ ∀𝑑 ∈ 𝑏 𝜃) → 𝜑))    ⇒   ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂)
 
26-Aug-2024df-nadd 33752 Define natural ordinal addition. This is a commutative form of addition over the ordinals. (Contributed by Scott Fenton, 26-Aug-2024.)
+no = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), (𝑧 ∈ V, 𝑎 ∈ V ↦ ∩ {𝑤 ∈ On ∣ ((𝑎 “ ({(1st ‘𝑧)} × (2nd ‘𝑧))) ⊆ 𝑤 ∧ (𝑎 “ ((1st ‘𝑧) × {(2nd ‘𝑧)})) ⊆ 𝑤)}))
 
26-Aug-2024findcard2 8909 Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.) Avoid ax-pow 5283. (Revised by BTernaryTau, 26-Aug-2024.)
(𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   𝜓    &   (𝑦 ∈ Fin → (𝜒 → 𝜃))    ⇒   (𝐴 ∈ Fin → 𝜏)
 
26-Aug-2024dif1en 8907 If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. For a proof with fewer symbols using ax-pow 5283, see dif1enALT 8980. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5283. (Revised by BTernaryTau, 26-Aug-2024.)
((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
 
26-Aug-2024rexdif1en 8906 If a set is equinumerous to a nonzero finite ordinal, then there exists an element in that set such that removing it leaves the set equinumerous to the predecessor of that ordinal. (Contributed by BTernaryTau, 26-Aug-2024.)
((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
 
25-Aug-2024flt4lem7 40412 Convert flt4lem5f 40410 into a convenient form for nna4b4nsq 40413. TODO-SN: The change to (𝐴 gcd 𝐵) = 1 points at some inefficiency in the lemmas. (Contributed by SN, 25-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → ∃𝑙 ∈ ℕ (∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑙↑2))) ∧ 𝑙 < 𝐶))
 
25-Aug-2024gcdle2d 40252 The greatest common divisor of a positive integer and another integer is less than or equal to the positive integer. (Contributed by SN, 25-Aug-2024.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → (𝑀 gcd 𝑁) ≤ 𝑁)
 
25-Aug-2024gcdle1d 40251 The greatest common divisor of a positive integer and another integer is less than or equal to the positive integer. (Contributed by SN, 25-Aug-2024.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝑀 gcd 𝑁) ≤ 𝑀)
 
25-Aug-2024gcdnn0id 40250 The gcd of a nonnegative integer and itself is the integer. (Contributed by SN, 25-Aug-2024.)
(𝑁 ∈ ℕ0 → (𝑁 gcd 𝑁) = 𝑁)
 
24-Aug-2024flt4lem5f 40410 Final equation of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. Given 𝐴↑4 + 𝐵↑4 = 𝐶↑2, provide a smaller solution. This satisfies the infinite descent condition. (Contributed by SN, 24-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → ((𝑀 gcd (𝐵 / 2))↑2) = (((𝑅 gcd (𝐵 / 2))↑4) + ((𝑆 gcd (𝐵 / 2))↑4)))
 
24-Aug-2024flt4lem5elem 40404 Version of fltaccoprm 40393 and fltbccoprm 40394 where 𝑀 is not squared. This can be proved in general for any polynomial in three variables: using prmdvdsncoprmbd 16359, dvds2addd 15929, and prmdvdsexp 16348, we can show that if two variables are coprime, the third is also coprime to the two. (Contributed by SN, 24-Aug-2024.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑆 ∈ ℕ)    &   (𝜑 → 𝑀 = ((𝑅↑2) + (𝑆↑2)))    &   (𝜑 → (𝑅 gcd 𝑆) = 1)    ⇒   (𝜑 → ((𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1))
 
24-Aug-20242rspcedvdw 40108 Double application of rspcedvdw 40107. (Contributed by SN, 24-Aug-2024.)
(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))    &   (𝜑 → 𝐴 ∈ 𝑋)    &   (𝜑 → 𝐵 ∈ 𝑌)    &   (𝜑 → 𝜃)    ⇒   (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓)
 
24-Aug-2024brttrcl2 33700 Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 24-Aug-2024.)
(𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ ω ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
 
24-Aug-2024prmdvdsncoprmbd 16359 Two positive integers are not coprime iff a prime divides both integers. Deduction version of ncoprmgcdne1b 16283 with the existential quantifier over the primes instead of integers greater than or equal to 2. (Contributed by SN, 24-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    ⇒   (𝜑 → (∃𝑝 ∈ ℙ (𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) ↔ (𝐴 gcd 𝐵) ≠ 1))
 
24-Aug-2024gcdcomd 16149 The gcd operator is commutative, deduction version. (Contributed by SN, 24-Aug-2024.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀))
 
23-Aug-2024nna4b4nsq 40413 Strengthening of Fermat's last theorem for exponent 4, where the sum is only assumed to be a square. (Contributed by SN, 23-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    ⇒   (𝜑 → ((𝐴↑4) + (𝐵↑4)) ≠ (𝐶↑2))
 
23-Aug-2024flt4lem5e 40409 Satisfy the hypotheses of flt4lem4 40402. (Contributed by SN, 23-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → (((𝑅 gcd 𝑆) = 1 ∧ (𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1) ∧ (𝑅 ∈ ℕ ∧ 𝑆 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ ((𝑀 · (𝑅 · 𝑆)) = ((𝐵 / 2)↑2) ∧ (𝐵 / 2) ∈ ℕ)))
 
23-Aug-2024flt4lem5d 40408 Part 3 of Equation 2 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 23-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → 𝑀 = ((𝑅↑2) + (𝑆↑2)))
 
23-Aug-2024flt4lem5 40403 In the context of the lemmas of pythagtrip 16463, 𝑀 and 𝑁 are coprime. (Contributed by SN, 23-Aug-2024.)
𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)    &   𝑁 = (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)    ⇒   (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑀 gcd 𝑁) = 1)
 
23-Aug-2024posqsqznn 40264 When a positive rational squared is an integer, the rational is a positive integer. zsqrtelqelz 16390 with all terms squared and positive. (Contributed by SN, 23-Aug-2024.)
(𝜑 → (𝐴↑2) ∈ ℤ)    &   (𝜑 → 𝐴 ∈ ℚ)    &   (𝜑 → 0 < 𝐴)    ⇒   (𝜑 → 𝐴 ∈ ℕ)
 
23-Aug-2024lsubcom23d 40228 Swap the second and third variables in an equation with subtraction on the left, converting it into an addition. (Contributed by SN, 23-Aug-2024.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 − 𝐵) = 𝐶)    ⇒   (𝜑 → (𝐴 − 𝐶) = 𝐵)
 
23-Aug-2024addscllem1 34058 Lemma for addscl (future) Alternate expression for surreal addition. (Contributed by Scott Fenton, 23-Aug-2024.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = ((( +s “ (( L ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( L ‘𝐵)))) |s (( +s “ (( R ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( R ‘𝐵))))))
 
23-Aug-2024scutfo 34011 The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.)
|s : <<s –onto→ No
 
23-Aug-2024scutcld 33924 Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
(𝜑 → 𝐴 <<s 𝐵)    ⇒   (𝜑 → (𝐴 |s 𝐵) ∈ No )
 
23-Aug-2024scutcl 33923 Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
(𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
 
23-Aug-2024dvdsmultr2d 15936 Deduction form of dvdsmultr2 15935. (Contributed by SN, 23-Aug-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ∥ 𝑁)    ⇒   (𝜑 → 𝐾 ∥ (𝑀 · 𝑁))
 
23-Aug-2024dmfexALT 7731 Alternate proof of dmfex 7728: shorter but using ax-rep 5205. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Proof shortened by AV, 23-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
 
23-Aug-2024nfceqdf 2901 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) Avoid ax-8 2110 and df-clel 2817. (Revised by WL and SN, 23-Aug-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))
 
23-Aug-2024eleq2w2 2734 A weaker version of eleq2 2827 (but stronger than ax-9 2118 and elequ2 2123) that uses ax-12 2173 to avoid ax-8 2110 and df-clel 2817. Compare eleq2w 2822, whose setvars appear where the class variables are in this theorem, and vice versa. (Contributed by BJ, 24-Jun-2019.) Strengthen from setvar variables to class variables. (Revised by WL and SN, 23-Aug-2024.)
(𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
 
22-Aug-2024flt4lem5c 40407 Part 2 of Equation 2 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 22-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → 𝑁 = (2 · (𝑅 · 𝑆)))
 
22-Aug-2024flt4lem5b 40406 Part 2 of Equation 1 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 22-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → (2 · (𝑀 · 𝑁)) = (𝐵↑2))
 
22-Aug-2024flt4lem5a 40405 Part 1 of Equation 1 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 22-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2))
 
22-Aug-2024flt4lem4 40402 If the product of two coprime factors is a perfect square, the factors are perfect squares. (Contributed by SN, 22-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    &   (𝜑 → (𝐴 · 𝐵) = (𝐶↑2))    ⇒   (𝜑 → (𝐴 = ((𝐴 gcd 𝐶)↑2) ∧ 𝐵 = ((𝐵 gcd 𝐶)↑2)))
 
22-Aug-2024flt4lem3 40401 Equivalent to pythagtriplem4 16448. Show that 𝐶 + 𝐴 and 𝐶 − 𝐴 are coprime. (Contributed by SN, 22-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   (𝜑 → ((𝐶 + 𝐴) gcd (𝐶 − 𝐴)) = 1)
 
22-Aug-2024flt4lem2 40400 If 𝐴 is even, 𝐵 is odd. (Contributed by SN, 22-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   (𝜑 → ¬ 2 ∥ 𝐵)
 
22-Aug-2024fltabcoprm 40395 A counterexample to FLT with 𝐴, 𝐶 coprime also has 𝐴, 𝐵 coprime. Converse of fltaccoprm 40393. (Contributed by SN, 22-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   (𝜑 → (𝐴 gcd 𝐵) = 1)
 
22-Aug-2024no2inds 34039 Double induction on surreals. The many substitution instances are to cover all possible cases. (Contributed by Scott Fenton, 22-Aug-2024.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))    &   (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))    &   ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃) → 𝜑))    ⇒   ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂)
 
22-Aug-2024no2indslem 34038 Double induction on surreals with explicit notation for the relationships. (Contributed by Scott Fenton, 22-Aug-2024.)
𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   𝑆 = {⟨𝑐, 𝑑⟩ ∣ (𝑐 ∈ ( No × No ) ∧ 𝑑 ∈ ( No × No ) ∧ (((1st ‘𝑐)𝑅(1st ‘𝑑) ∨ (1st ‘𝑐) = (1st ‘𝑑)) ∧ ((2nd ‘𝑐)𝑅(2nd ‘𝑑) ∨ (2nd ‘𝑐) = (2nd ‘𝑑)) ∧ 𝑐 ≠ 𝑑))}    &   (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))    &   (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))    &   ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃) → 𝜑))    ⇒   ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂)
 
22-Aug-2024xpord2ind 33721 Induction over the cross product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 22-Aug-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   𝑅 Fr 𝐴    &   𝑅 Po 𝐴    &   𝑅 Se 𝐴    &   𝑆 Fr 𝐵    &   𝑆 Po 𝐵    &   𝑆 Se 𝐵    &   (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))    &   (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))    &   (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))    &   (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))    &   (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))    &   ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑))    ⇒   ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)
 
22-Aug-2024xpord2pred 33719 Calculate the predecessor class in frxp2 33718. (Contributed by Scott Fenton, 22-Aug-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑌⟩) = (((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) ∖ {⟨𝑋, 𝑌⟩}))
 
22-Aug-2024frpoins3xp3g 33715 Special case of founded partial recursion over a triple cross product. (Contributed by Scott Fenton, 22-Aug-2024.)
((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (∀𝑤∀𝑡∀𝑢(⟨⟨𝑤, 𝑡⟩, 𝑢⟩ ∈ Pred(𝑅, ((𝐴 × 𝐵) × 𝐶), ⟨⟨𝑥, 𝑦⟩, 𝑧⟩) → 𝜃) → 𝜑))    &   (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝑡 → (𝜓 ↔ 𝜒))    &   (𝑧 = 𝑢 → (𝜒 ↔ 𝜃))    &   (𝑥 = 𝑋 → (𝜑 ↔ 𝜏))    &   (𝑦 = 𝑌 → (𝜏 ↔ 𝜂))    &   (𝑧 = 𝑍 → (𝜂 ↔ 𝜁))    ⇒   (((𝑅 Fr ((𝐴 × 𝐵) × 𝐶) ∧ 𝑅 Po ((𝐴 × 𝐵) × 𝐶) ∧ 𝑅 Se ((𝐴 × 𝐵) × 𝐶)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶)) → 𝜁)
 
22-Aug-2024frpoins3xpg 33714 Special case of founded partial induction over a cross product. (Contributed by Scott Fenton, 22-Aug-2024.)
((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒) → 𝜑))    &   (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))    &   (𝑥 = 𝑋 → (𝜑 ↔ 𝜃))    &   (𝑦 = 𝑌 → (𝜃 ↔ 𝜏))    ⇒   (((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝜏)
 
22-Aug-2024ralxp3es 33591 Restricted for-all over a triple cross product with explicit substitution. (Contributed by Scott Fenton, 22-Aug-2024.)
(∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)[(1st ‘(1st ‘𝑥)) / 𝑦][(2nd ‘(1st ‘𝑥)) / 𝑧][(2nd ‘𝑥) / 𝑤]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜑)
 
22-Aug-2024sbcoteq1a 33590 Equality theorem for substitution of a class for an ordered triple. (Contributed by Scott Fenton, 22-Aug-2024.)
(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ([(1st ‘(1st ‘𝐴)) / 𝑥][(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑 ↔ 𝜑))
 
22-Aug-2024ralxp3f 33588 Restricted for all over a triple cross product. (Contributed by Scott Fenton, 22-Aug-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑧𝜑    &   Ⅎ𝑤𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
 
22-Aug-2024f1osetex 8605 The set of bijections between two classes exists. (Contributed by AV, 30-Mar-2024.) (Revised by AV, 8-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
{𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ∈ V
 
22-Aug-2024fsetdmprc0 8601 The set of functions with a proper class as domain is empty. (Contributed by AV, 22-Aug-2024.)
(𝐴 ∉ V → {𝑓 ∣ 𝑓 Fn 𝐴} = ∅)
 
22-Aug-2024abanssr 4233 A class abstraction with a conjunction is a subset of the class abstraction with the right conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
{𝑓 ∣ (𝜑 ∧ 𝜓)} ⊆ {𝑓 ∣ 𝜓}
 
22-Aug-2024abanssl 4232 A class abstraction with a conjunction is a subset of the class abstraction with the left conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
{𝑓 ∣ (𝜑 ∧ 𝜓)} ⊆ {𝑓 ∣ 𝜑}
 
21-Aug-2024flt4lem1 40399 Satisfy the antecedent used in several pythagtrip 16463 lemmas, with 𝐴, 𝐶 coprime rather than 𝐴, 𝐵. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   (𝜑 → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)))
 
21-Aug-2024flt4lem 40398 Raising a number to the fourth power is equivalent to squaring it twice. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐴 ∈ ℂ)    ⇒   (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2))
 
21-Aug-2024fltdvdsabdvdsc 40391 Any factor of both 𝐴 and 𝐵 also divides 𝐶. This establishes the validity of fltabcoprmex 40392. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶)
 
21-Aug-2024dvdsexpad 40253 Deduction associated with dvdsexpim 40249. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐴 ∈ ℤ)    &   (𝜑 → 𝐵 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐴 ∥ 𝐵)    ⇒   (𝜑 → (𝐴↑𝑁) ∥ (𝐵↑𝑁))
 
21-Aug-2024lsubrotld 40227 Rotate the variables left in an equation with subtraction on the left, converting it into an addition. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 − 𝐵) = 𝐶)    ⇒   (𝜑 → (𝐵 + 𝐶) = 𝐴)
 
21-Aug-2024raddcom12d 40226 Swap the first two variables in an equation with addition on the right, converting it into a subtraction. Version of mvrraddd 11317 with a commuted consequent, and of mvlraddd 11315 with a commuted hypothesis. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐴 = (𝐵 + 𝐶))    ⇒   (𝜑 → 𝐵 = (𝐴 − 𝐶))
 
21-Aug-2024laddrotrd 40225 Rotate the variables right in an equation with addition on the left, converting it into a subtraction. Version of mvlladdd 11316 with a commuted consequent, and of mvrladdd 11318 with a commuted hypothesis. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) = 𝐶)    ⇒   (𝜑 → (𝐶 − 𝐴) = 𝐵)
 
21-Aug-2024mvrrsubd 40224 Move a subtraction in the RHS to a right-addition in the LHS. Converse of mvlraddd 11315. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐴 = (𝐵 − 𝐶))    ⇒   (𝜑 → (𝐴 + 𝐶) = 𝐵)
 
21-Aug-2024aks4d1p1 40012 Show inequality for existence of a non-divisor. (Contributed by metakunt, 21-Aug-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   (𝜑 → 𝐴 < (2↑𝐵))
 
21-Aug-20243lexlogpow5ineq5 39996 Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.)
((2 logb 3)↑5) ≤ 15
 
21-Aug-20243lexlogpow2ineq2 39995 Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.)
(2 < ((2 logb 3)↑2) ∧ ((2 logb 3)↑2) < 3)
 
21-Aug-20243lexlogpow2ineq1 39994 Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.)
((3 / 2) < (2 logb 3) ∧ (2 logb 3) < (5 / 3))
 
21-Aug-20243lexlogpow5ineq4 39992 Sharper logarithm inequality chain. (Contributed by metakunt, 21-Aug-2024.)
(𝜑 → 𝑋 ∈ ℝ)    &   (𝜑 → 3 ≤ 𝑋)    ⇒   (𝜑 → 9 < ((2 logb 𝑋)↑5))
 
21-Aug-20243exp7 39989 3 to the power of 7 equals 2187. (Contributed by metakunt, 21-Aug-2024.)
(3↑7) = 2187
 
21-Aug-2024sexp3 33726 Show that the triple order is set-like. (Contributed by Scott Fenton, 21-Aug-2024.)
𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   (𝜑 → 𝑅 Se 𝐴)    &   (𝜑 → 𝑆 Se 𝐵)    &   (𝜑 → 𝑇 Se 𝐶)    ⇒   (𝜑 → 𝑈 Se ((𝐴 × 𝐵) × 𝐶))
 
21-Aug-2024xpord3pred 33725 Calculate the predecsessor class for the triple order. (Contributed by Scott Fenton, 21-Aug-2024.)
𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    ⇒   ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨⟨𝑋, 𝑌⟩, 𝑍⟩) = ((((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) × (Pred(𝑇, 𝐶, 𝑍) ∪ {𝑍})) ∖ {⟨⟨𝑋, 𝑌⟩, 𝑍⟩}))
 
21-Aug-2024frxp3 33724 Give foundedness over a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.)
𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   (𝜑 → 𝑅 Fr 𝐴)    &   (𝜑 → 𝑆 Fr 𝐵)    &   (𝜑 → 𝑇 Fr 𝐶)    ⇒   (𝜑 → 𝑈 Fr ((𝐴 × 𝐵) × 𝐶))
 
21-Aug-2024poxp3 33723 Triple cross product partial ordering. (Contributed by Scott Fenton, 21-Aug-2024.)
𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   (𝜑 → 𝑅 Po 𝐴)    &   (𝜑 → 𝑆 Po 𝐵)    &   (𝜑 → 𝑇 Po 𝐶)    ⇒   (𝜑 → 𝑈 Po ((𝐴 × 𝐵) × 𝐶))
 
21-Aug-2024xpord3lem 33722 Lemma for triple ordering. Calculate the value of the relationship. (Contributed by Scott Fenton, 21-Aug-2024.)
𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    ⇒   (⟨⟨𝑎, 𝑏⟩, 𝑐⟩𝑈⟨⟨𝑑, 𝑒⟩, 𝑓⟩ ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶) ∧ (((𝑎𝑅𝑑 ∨ 𝑎 = 𝑑) ∧ (𝑏𝑆𝑒 ∨ 𝑏 = 𝑒) ∧ (𝑐𝑇𝑓 ∨ 𝑐 = 𝑓)) ∧ (𝑎 ≠ 𝑑 ∨ 𝑏 ≠ 𝑒 ∨ 𝑐 ≠ 𝑓))))
 
21-Aug-2024ralxp3 33589 Restricted for-all over a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.)
(𝑝 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   (∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)
 
21-Aug-2024elxpxpss 33587 Version of elrel 5697 for triple cross products. (Contributed by Scott Fenton, 21-Aug-2024.)
((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
 
21-Aug-2024elxpxp 33586 Membership in a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.)
(𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
 
21-Aug-2024otthne 33585 Contrapositive of the ordered triple theorem. (Contributed by Scott Fenton, 21-Aug-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    ⇒   (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ≠ ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹))
 
21-Aug-2024ot22ndd 33584 Extract the second member of an ordered triple. Deduction version. (Contributed by Scott Fenton, 21-Aug-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    ⇒   (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (2nd ‘(1st ‘𝑋)) = 𝐵)
 
21-Aug-2024ot21std 33583 Extract the first member of an ordered triple. Deduction version. (Contributed by Scott Fenton, 21-Aug-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    ⇒   (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (1st ‘(1st ‘𝑋)) = 𝐴)
 
21-Aug-2024ot2elxp 33582 Ordered triple membership in a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.)
(⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹))
 
21-Aug-2024ralxpes 33581 A version of ralxp 5739 with explicit substitution. (Contributed by Scott Fenton, 21-Aug-2024.)
(∀𝑥 ∈ (𝐴 × 𝐵)[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜑)
 
21-Aug-2024prmdvdssq 16351 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by SN, 21-Aug-2024.)
((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ) → (𝑃 ∥ 𝑀 ↔ 𝑃 ∥ (𝑀↑2)))
 
21-Aug-2024rprpwr 16196 If 𝐴 and 𝐵 are relatively prime, then so are 𝐴 and 𝐵↑𝑁. Originally a subproof of rppwr 16197. (Contributed by SN, 21-Aug-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → (𝐴 gcd (𝐵↑𝑁)) = 1))
 
21-Aug-2024dvds2addd 15929 Deduction form of dvds2add 15927. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ∥ 𝑀)    &   (𝜑 → 𝐾 ∥ 𝑁)    ⇒   (𝜑 → 𝐾 ∥ (𝑀 + 𝑁))
 
20-Aug-2024infdesc 40396 Infinite descent. The hypotheses say that 𝑆 is lower bounded, and that if 𝜓 holds for an integer in 𝑆, it holds for a smaller integer in 𝑆. By infinite descent, eventually we cannot go any smaller, therefore 𝜓 holds for no integer in 𝑆. (Contributed by SN, 20-Aug-2024.)
(𝑦 = 𝑥 → (𝜓 ↔ 𝜒))    &   (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))    &   (𝜑 → 𝑆 ⊆ (ℤ≥‘𝑀))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝜒)) → ∃𝑧 ∈ 𝑆 (𝜃 ∧ 𝑧 < 𝑥))    ⇒   (𝜑 → {𝑦 ∈ 𝑆 ∣ 𝜓} = ∅)
 
20-Aug-2024fltbccoprm 40394 A counterexample to FLT with 𝐴, 𝐵 coprime also has 𝐵, 𝐶 coprime. Proven from fltaccoprm 40393 using commutativity of addition. (Contributed by SN, 20-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    ⇒   (𝜑 → (𝐵 gcd 𝐶) = 1)
 
20-Aug-2024fltaccoprm 40393 A counterexample to FLT with 𝐴, 𝐵 coprime also has 𝐴, 𝐶 coprime. (Contributed by SN, 20-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    ⇒   (𝜑 → (𝐴 gcd 𝐶) = 1)
 
20-Aug-2024fltabcoprmex 40392 A counterexample to FLT implies a counterexample to FLT with 𝐴, 𝐵 (assigned to 𝐴 / (𝐴 gcd 𝐵) and 𝐵 / (𝐴 gcd 𝐵)) coprime (by divgcdcoprm0 16298). (Contributed by SN, 20-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   (𝜑 → (((𝐴 / (𝐴 gcd 𝐵))↑𝑁) + ((𝐵 / (𝐴 gcd 𝐵))↑𝑁)) = ((𝐶 / (𝐴 gcd 𝐵))↑𝑁))
 
20-Aug-2024flt0 40390 A counterexample for FLT does not exist for 𝑁 = 0. (Contributed by SN, 20-Aug-2024.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   (𝜑 → 𝑁 ∈ ℕ)
 
20-Aug-2024fltdiv 40389 A counterexample to FLT stays valid when scaled. The hypotheses are more general than they need to be for convenience. (Contributed by SN, 20-Aug-2024.)
(𝜑 → 𝑆 ∈ ℂ)    &   (𝜑 → 𝑆 ≠ 0)    &   (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   (𝜑 → (((𝐴 / 𝑆)↑𝑁) + ((𝐵 / 𝑆)↑𝑁)) = ((𝐶 / 𝑆)↑𝑁))
 
20-Aug-2024fltmul 40388 A counterexample to FLT stays valid when scaled. The hypotheses are more general than they need to be for convenience. (There does not seem to be a standard term for Fermat or Pythagorean triples extended to any 𝑁 ∈ ℕ0, so the label is more about the context in which this theorem is used). (Contributed by SN, 20-Aug-2024.)
(𝜑 → 𝑆 ∈ ℂ)    &   (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   (𝜑 → (((𝑆 · 𝐴)↑𝑁) + ((𝑆 · 𝐵)↑𝑁)) = ((𝑆 · 𝐶)↑𝑁))
 
20-Aug-2024dvdsexpnn 40261 dvdssqlem 16199 generalized to positive integer exponents. (Contributed by SN, 20-Aug-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 ∥ 𝐵 ↔ (𝐴↑𝑁) ∥ (𝐵↑𝑁)))
 
20-Aug-20243rspcedvdw 40109 Triple application of rspcedvdw 40107. (Contributed by SN, 20-Aug-2024.)
(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))    &   (𝑧 = 𝐶 → (𝜃 ↔ 𝜏))    &   (𝜑 → 𝐴 ∈ 𝑋)    &   (𝜑 → 𝐵 ∈ 𝑌)    &   (𝜑 → 𝐶 ∈ 𝑍)    &   (𝜑 → 𝜏)    ⇒   (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∃𝑧 ∈ 𝑍 𝜓)
 
20-Aug-2024rspcedvdw 40107 Version of rspcedvd 3555 where the implicit substitution hypothesis does not have an antecedent, which also avoids a disjoint variable condition on 𝜑, 𝑥. (Contributed by SN, 20-Aug-2024.)
(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   (𝜑 → 𝐴 ∈ 𝐵)    &   (𝜑 → 𝜒)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
 
20-Aug-2024addscomd 34057 Surreal addition commutes. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))
 
20-Aug-2024addscom 34056 Surreal addition commutes. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))
 
20-Aug-2024addsid1d 34055 Surreal addition to zero is identity. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
(𝜑 → 𝐴 ∈ No )    ⇒   (𝜑 → (𝐴 +s 0s ) = 𝐴)
 
20-Aug-2024addsid1 34054 Surreal addition to zero is identity. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
(𝐴 ∈ No → (𝐴 +s 0s ) = 𝐴)
 
20-Aug-2024addsval 34053 The value of surreal addition. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑦 = (𝑟 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝐵)𝑧 = (𝐴 +s 𝑟)})))
 
20-Aug-2024addsfn 34052 Surreal addition is a function over pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
+s Fn ( No × No )
 
20-Aug-2024negs0s 34051 Negative surreal zero is surreal zero. (Contributed by Scott Fenton, 20-Aug-2024.)
( -us ‘ 0s ) = 0s
 
20-Aug-2024negsval 34050 The value of the surreal negation function. (Contributed by Scott Fenton, 20-Aug-2024.)
(𝐴 ∈ No → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
 
20-Aug-2024negsfn 34049 Surreal negation is a function over surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
-us Fn No
 
20-Aug-2024df-subs 34048 Define surreal subtraction. (Contributed by Scott Fenton, 20-Aug-2024.)
-s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦)))
 
20-Aug-2024df-negs 34047 Define surreal negation. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.)
-us = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))))
 
20-Aug-2024df-adds 34046 Define surreal addition. This is the first of the field operations on the surreals. Definition from [Conway] p. 5. Definition from [Gonshor] p. 13. (Contributed by Scott Fenton, 20-Aug-2024.)
+s = norec2 ((𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ ( L ‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ ( R ‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)}))))
 
20-Aug-2024norec2ov 34041 The value of the double-recursion surreal function. (Contributed by Scott Fenton, 20-Aug-2024.)
𝐹 = norec2 (𝐺)    ⇒   ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))))
 
20-Aug-2024norec2fn 34040 The double-recursion operator on surreals yields a function on pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
𝐹 = norec2 (𝐺)    ⇒   𝐹 Fn ( No × No )
 
20-Aug-2024df-norec2 34033 Define surreal recursion on two variables. This function is key to the development of most of surreal numbers. (Contributed by Scott Fenton, 20-Aug-2024.)
norec2 (𝐹) = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), 𝐹)
 
20-Aug-2024right0s 34003 The right set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
( R ‘ 0s ) = ∅
 
20-Aug-2024left0s 34002 The left set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
( L ‘ 0s ) = ∅
 
20-Aug-2024ssltsepc 33914 Two elements of separated sets obey less than. (Contributed by Scott Fenton, 20-Aug-2024.)
((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌)
 
19-Aug-2024aks4d1p1p5 40011 Show inequality for existence of a non-divisor. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝑁 ∈ ℕ)    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   (𝜑 → 4 ≤ 𝑁)    &   𝐶 = (2 logb (((2 logb 𝑁)↑5) + 1))    &   𝐷 = ((2 logb 𝑁)↑2)    &   𝐸 = ((2 logb 𝑁)↑4)    ⇒   (𝜑 → 𝐴 < (2↑𝐵))
 
19-Aug-2024aks4d1p1p7 40010 Bound of intermediary of inequality step. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 4 ≤ 𝐴)    ⇒   (𝜑 → ((2 · ((1 / ((((2 logb 𝐴)↑5) + 1) · (log‘2))) · (((5 · ((2 logb 𝐴)↑4)) · (1 / (𝐴 · (log‘2)))) + 0))) + ((2 / ((log‘2)↑2)) · (((log‘𝐴)↑(2 − 1)) / 𝐴))) ≤ ((4 / ((log‘2)↑4)) · (((log‘𝐴)↑3) / 𝐴)))
 
19-Aug-2024aks4d1p1p6 40009 Inequality lift to differentiable functions for a term in AKS inequality lemma. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 3 ≤ 𝐴)    &   (𝜑 → 𝐴 ≤ 𝐵)    ⇒   (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · (2 logb (((2 logb 𝑥)↑5) + 1))) + ((2 logb 𝑥)↑2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · ((1 / ((((2 logb 𝑥)↑5) + 1) · (log‘2))) · (((5 · ((2 logb 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0))) + ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥)))))
 
19-Aug-2024dvle2 40008 Collapsed dvle 25076. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐸)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐹))    &   (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐺)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐻))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐹 ≤ 𝐻)    &   (𝑥 = 𝐴 → 𝐸 = 𝑃)    &   (𝑥 = 𝐴 → 𝐺 = 𝑄)    &   (𝑥 = 𝐵 → 𝐸 = 𝑅)    &   (𝑥 = 𝐵 → 𝐺 = 𝑆)    &   (𝜑 → 𝑃 ≤ 𝑄)    &   (𝜑 → 𝐴 ≤ 𝐵)    ⇒   (𝜑 → 𝑅 ≤ 𝑆)
 
19-Aug-2024aks4d1p1p4 40007 Technical step for inequality. The hard work is in to prove the final hypothesis. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝑁 ∈ ℕ)    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   (𝜑 → 3 ≤ 𝑁)    &   𝐶 = (2 logb (((2 logb 𝑁)↑5) + 1))    &   𝐷 = ((2 logb 𝑁)↑2)    &   𝐸 = ((2 logb 𝑁)↑4)    &   (𝜑 → ((2 · 𝐶) + 𝐷) ≤ 𝐸)    ⇒   (𝜑 → 𝐴 < (2↑𝐵))
 
19-Aug-2024aks4d1p1p2 40006 Rewrite 𝐴 in more suitable form. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝑁 ∈ ℕ)    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   (𝜑 → 3 ≤ 𝑁)    ⇒   (𝜑 → 𝐴 < (𝑁↑𝑐(((2 logb (((2 logb 𝑁)↑5) + 1)) + (((2 logb 𝑁)↑2) / 2)) + (((2 logb 𝑁)↑4) / 2))))
 
19-Aug-2024aks4d1p1p3 40005 Bound of a ceiling of the binary logarithm to the fifth power. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝑁 ∈ ℕ)    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   (𝜑 → 3 ≤ 𝑁)    ⇒   (𝜑 → (𝑁↑𝑐(⌊‘(2 logb 𝐵))) < (𝑁↑𝑐(2 logb (((2 logb 𝑁)↑5) + 1))))
 
19-Aug-2024noxpordpred 34037 Next we calculate the predecessor class of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑆, ( No × No ), ⟨𝐴, 𝐵⟩) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
 
19-Aug-2024noxpordse 34036 Next we establish the set-like nature of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   𝑆 Se ( No × No )
 
19-Aug-2024noxpordfr 34035 Next we establish the foundedness of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   𝑆 Fr ( No × No )
 
19-Aug-2024noxpordpo 34034 To get through most of the textbook defintions in surreal numbers we will need recursion on two variables. This set of theorems sets up the preconditions for double recursion. This theorem establishes the partial ordering. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   𝑆 Po ( No × No )
 
19-Aug-2024norecov 34031 Calculate the value of the surreal recursion operation. (Contributed by Scott Fenton, 19-Aug-2024.)
𝐹 = norec (𝐺)    ⇒   (𝐴 ∈ No → (𝐹‘𝐴) = (𝐴𝐺(𝐹 ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))))
 
19-Aug-2024norecfn 34030 Surreal recursion over one variable is a function over the surreals. (Contributed by Scott Fenton, 19-Aug-2024.)
𝐹 = norec (𝐺)    ⇒   𝐹 Fn No
 
19-Aug-2024noinds 34029 Induction principle for a single surreal. If a property passes from a surreal's left and right sets to the surreal itself, then it holds for all surreals. (Contributed by Scott Fenton, 19-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   (𝑥 ∈ No → (∀𝑦 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 → 𝜑))    ⇒   (𝐴 ∈ No → 𝜒)
 
19-Aug-2024lrrecpred 34028 Finally, we calculate the value of the predecessor class over 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
 
19-Aug-2024lrrecfr 34027 Now we show that 𝑅 is founded over No . (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   𝑅 Fr No
 
19-Aug-2024lrrecse 34026 Next, we show that 𝑅 is set-like over No . (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   𝑅 Se No
 
19-Aug-2024lrrecpo 34025 Now, we establish that 𝑅 is a partial ordering on No . (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   𝑅 Po No
 
19-Aug-2024lrrecval2 34024 Next, we establish an alternate expression for 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
 
19-Aug-2024lrrecval 34023 The next step in the development of the surreals is to establish induction and recursion across left and right sets. To that end, we are going to develop a relationship 𝑅 that is founded, partial, and set-like across the surreals. This first theorem just establishes the value of 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))))
 
19-Aug-2024df-norec 34022 Define the recursion generator for surreal functions of one variable. This generator creates a recursive function of surreals from their value on their left and right sets. (Contributed by Scott Fenton, 19-Aug-2024.)
norec (𝐹) = frecs({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐹)
 
19-Aug-2024lrcut 34010 A surreal is equal to the cut of its left and right sets. (Contributed by Scott Fenton, 19-Aug-2024.)
(𝑋 ∈ No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
 
19-Aug-2024newbday 34009 A surreal is an element of a new set iff its birthday is equal to that ordinal. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday ‘𝑋) = 𝐴))
 
19-Aug-2024oldbday 34008 A surreal is part of the set older than ordinal 𝐴 iff its birthday is less than 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴))
 
19-Aug-2024madebday 34007 A surreal is part of the set made by ordinal 𝐴 iff its birthday is less than or equal to 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))
 
19-Aug-2024madebdaylemlrcut 34006 Lemma for madebday 34007. If the inductive hypothesis of madebday 34007 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 34010 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
 
19-Aug-2024madebdaylemold 34005 Lemma for madebday 34007. If the inductive hypothesis of madebday 34007 is satisfied, the converse of oldbdayim 33998 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴)))
 
19-Aug-2024sexp2 33720 Condition for the relationship in frxp2 33718 to be set-like. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   (𝜑 → 𝑅 Se 𝐴)    &   (𝜑 → 𝑆 Se 𝐵)    ⇒   (𝜑 → 𝑇 Se (𝐴 × 𝐵))
 
19-Aug-2024frxp2 33718 Another way of giving a founded order to a cross product of two classes. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   (𝜑 → 𝑅 Fr 𝐴)    &   (𝜑 → 𝑆 Fr 𝐵)    ⇒   (𝜑 → 𝑇 Fr (𝐴 × 𝐵))
 
19-Aug-2024poxp2 33717 Another way of partially ordering a cross product of two classes. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   (𝜑 → 𝑅 Po 𝐴)    &   (𝜑 → 𝑆 Po 𝐵)    ⇒   (𝜑 → 𝑇 Po (𝐴 × 𝐵))
 
19-Aug-2024xpord2lem 33716 Lemma for cross product ordering. Calculate the value of the cross product relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ ((𝑎𝑅𝑐 ∨ 𝑎 = 𝑐) ∧ (𝑏𝑆𝑑 ∨ 𝑏 = 𝑑) ∧ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑑))))
 
19-Aug-2024dfse3 33580 Alternate definition of set-like relationships. (Contributed by Scott Fenton, 19-Aug-2024.)
(𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 Pred(𝑅, 𝐴, 𝑥) ∈ V)
 
19-Aug-2024xpab 33579 Cross product of two class abstractions. (Contributed by Scott Fenton, 19-Aug-2024.)
({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
 
19-Aug-2024frpoins3g 6234 Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 19-Aug-2024.)
(𝑥 ∈ 𝐴 → (∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑥)𝜓 → 𝜑))    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝐵 ∈ 𝐴) → 𝜒)
 
18-Aug-2024brttrcl 33699 Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 18-Aug-2024.)
(𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
 
18-Aug-2024prdsco 17096 Structure product composition operation. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐻 = (Hom ‘𝑃)    &    ∙ = (comp‘𝑃)    ⇒   (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))
 
18-Aug-2024prdshom 17095 Structure product hom-sets. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐻 = (Hom ‘𝑃)    ⇒   (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))
 
18-Aug-2024prdstset 17094 Structure product topology. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝑂 = (TopSet‘𝑃)    ⇒   (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
 
18-Aug-2024prdsds 17092 Structure product distance function. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐷 = (dist‘𝑃)    ⇒   (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )))
 
18-Aug-2024prdsle 17090 Structure product weak ordering. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    ≤ = (le‘𝑃)    ⇒   (𝜑 → ≤ = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})

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.)

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