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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-Dec-2024 at 5:21 AM ET.
Recent Additions to the Metamath Proof Explorer   Notes (last updated 7-Dec-2020 )
DateLabelDescription
Theorem
 
15-Dec-2024smfpimgtxr 44315 Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑 → 𝐴 ∈ ℝ*)    ⇒   (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
 
15-Dec-2024smfpimltxr 44283 Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑 → 𝐴 ∈ ℝ*)    ⇒   (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷))
 
15-Dec-2024pimltpnf2 44250 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)
 
15-Dec-2024pimltpnf2f 44249 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   Ⅎ𝑥𝐴    &   (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)
 
15-Dec-2024pimgtpnf2 44243 Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound +∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)
 
15-Dec-2024pimltmnf2 44236 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅)
 
15-Dec-2024pimltmnf2f 44235 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   Ⅎ𝑥𝐴    &   (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅)
 
14-Dec-2024fzuntgd 41065 Union of two adjacent or overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝐿 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ≤ 𝑀)    &   (𝜑 → 𝑀 ≤ (𝐿 + 1))    &   (𝜑 → 𝐿 ≤ 𝑁)    ⇒   (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁))
 
14-Dec-2024fzunt1d 41064 Union of two overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝐿 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ≤ 𝑀)    &   (𝜑 → 𝑀 ≤ 𝐿)    &   (𝜑 → 𝐿 ≤ 𝑁)    ⇒   (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁))
 
14-Dec-2024fzuntd 41063 Union of two adjacent finite sets of sequential integers that share a common endpoint. (Contributed by RP, 14-Dec-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ≤ 𝑀)    &   (𝜑 → 𝑀 ≤ 𝑁)    ⇒   (𝜑 → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))
 
14-Dec-2024fzunt 41062 Union of two adjacent finite sets of sequential integers that share a common endpoint. (Suggested by NM, 21-Jul-2005.) (Contributed by RP, 14-Dec-2024.)
(((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))
 
13-Dec-2024nlim4 41052 4 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
¬ Lim 4o
 
13-Dec-2024nlim3 41051 3 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
¬ Lim 3o
 
13-Dec-2024nlim2NEW 41050 2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
¬ Lim 2o
 
13-Dec-2024nlim1NEW 41049 1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
¬ Lim 1o
 
13-Dec-2024nlimsuc 41048 A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
(𝐴 ∈ On → ¬ Lim suc 𝐴)
 
13-Dec-2024wksonproplem 28072 Lemma for theorems for properties of walks between two vertices, e.g., trlsonprop 28076. (Contributed by AV, 16-Jan-2021.) Remove is-walk hypothesis. (Revised by SN, 13-Dec-2024.)
𝑉 = (Vtx‘𝐺)    &   (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃)))    &   𝑊 = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑂‘𝑔)𝑏)𝑝 ∧ 𝑓(𝑄‘𝑔)𝑝)}))    ⇒   (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃)))
 
13-Dec-2024mptmpoopabovd 7922 The operation value of a function value of a collection of ordered pairs of related elements. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable condition on 𝐷, 𝑓, ℎ to remove hypotheses. (Revised by SN, 13-Dec-2024.)
(𝜑 → 𝐺 ∈ 𝑊)    &   (𝜑 → 𝑋 ∈ (𝐴‘𝐺))    &   (𝜑 → 𝑌 ∈ (𝐵‘𝐺))    &   𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ∧ 𝑓(𝐷‘𝑔)ℎ)}))    ⇒   (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)})
 
13-Dec-2024mptmpoopabbrd 7921 The operation value of a function value of a collection of ordered pairs of elements related in two ways. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable condition on 𝐷, 𝑓, ℎ to remove hypotheses. (Revised by SN, 13-Dec-2024.)
(𝜑 → 𝐺 ∈ 𝑊)    &   (𝜑 → 𝑋 ∈ (𝐴‘𝐺))    &   (𝜑 → 𝑌 ∈ (𝐵‘𝐺))    &   ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝜏 ↔ 𝜃))    &   (𝑔 = 𝐺 → (𝜒 ↔ 𝜏))    &   𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)}))    ⇒   (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})
 
13-Dec-2024fvmptopab 7329 The function value of a mapping 𝑀 to a restricted binary relation expressed as an ordered-pair class abstraction: The restricted binary relation is a binary relation given as value of a function 𝐹 restricted by the condition 𝜓. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.) Add disjoint variable condition on 𝐹, 𝑥, 𝑦 to remove a sethood hypothesis. (Revised by SN, 13-Dec-2024.)
(𝑧 = 𝑍 → (𝜑 ↔ 𝜓))    &   𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜑)})    ⇒   (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}
 
13-Dec-2024opabresex2 7327 Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable conditions betweem 𝑊, 𝐺 and 𝑥, 𝑦 to remove hypotheses. (Revised by SN, 13-Dec-2024.)
{⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V
 
13-Dec-2024nfralw 3151 Bound-variable hypothesis builder for restricted quantification. Version of nfral 3153 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 13-Dec-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑
 
11-Dec-2024wksv 27986 The class of walks is a set. (Contributed by AV, 15-Jan-2021.) (Proof shortened by SN, 11-Dec-2024.)
{⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
 
11-Dec-2024abrexexg 7803 Existence of a class abstraction of existentially restricted sets. The class 𝐵 can be thought of as an expression in 𝑥 (which is typically a free variable in the class expression substituted for 𝐵) and the class abstraction appearing in the statement as the class of values 𝐵 as 𝑥 varies through 𝐴. If the "domain" 𝐴 is a set, then the abstraction is also a set. Therefore, this statement is a kind of Replacement. This can be seen by tracing back through the path axrep6g 5217, axrep6 5216, ax-rep 5209. See also abrexex2g 7807. There are partial converses under additional conditions, see for instance abnexg 7606. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2137, ax-11 2154, ax-12 2171, ax-pr 5352, ax-un 7588 and shorten proof. (Revised by SN, 11-Dec-2024.)
(𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
 
11-Dec-2024ssrel 5693 A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Remove dependency on ax-sep 5223, ax-nul 5230, ax-pr 5352. (Revised by KP, 25-Oct-2021.) Remove dependency on ax-12 2171. (Revised by SN, 11-Dec-2024.)
(Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
 
11-Dec-2024elopaelxp 5676 Membership in an ordered-pair class abstraction implies membership in a Cartesian product. (Contributed by Alexander van der Vekens, 23-Jun-2018.) Avoid ax-sep 5223, ax-nul 5230, ax-pr 5352. (Revised by SN, 11-Dec-2024.)
(𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))
 
11-Dec-2024elopabr 5474 Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) (Proof shortened by SN, 11-Dec-2024.)
(𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅)
 
11-Dec-2024elopabw 5439 Membership in a class abstraction of ordered pairs. Weaker version of elopab 5440 with a sethood antecedent, avoiding ax-sep 5223, ax-nul 5230, and ax-pr 5352. Originally a subproof of elopab 5440. (Contributed by SN, 11-Dec-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
 
11-Dec-2024axrep6g 5217 axrep6 5216 in class notation. It is equivalent to both ax-rep 5209 and abrexexg 7803, providing a direct link between the two. (Contributed by SN, 11-Dec-2024.)
((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)
 
11-Dec-2024dfiun2g 4960 Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Rohan Ridenour, 11-Aug-2023.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 11-Dec-2024.)
(∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
 
11-Dec-2024r19.21v 3113 Restricted quantifier version of 19.21v 1942. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020.) (Proof shortened by Wolf Lammen, 11-Dec-2024.)
(∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
 
10-Dec-2024sltn0 34085 If 𝑋 is less than 𝑌, then either ( L ‘𝑌) or ( R ‘𝑋) is non-empty. (Contributed by Scott Fenton, 10-Dec-2024.)
((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑋 <s 𝑌) → (( L ‘𝑌) ≠ ∅ ∨ ( R ‘𝑋) ≠ ∅))
 
10-Dec-2024cbvreuw 3376 Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3381 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-10 2137. (Revised by Wolf Lammen, 10-Dec-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
 
8-Dec-2024rexcom 3234 Commutation of restricted existential quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof shortened by BJ, 26-Aug-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2024.)
(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
 
5-Dec-2024sb8f 2351 Substitution of variable in universal quantifier. Version of sb8 2521 with a disjoint variable condition, not requiring ax-10 2137 or ax-13 2372. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) Avoid ax-10 2137. (Revised by SN, 5-Dec-2024.)
Ⅎ𝑦𝜑    ⇒   (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
 
5-Dec-2024sb8v 2350 Substitution of variable in universal quantifier. Version of sb8f 2351 with a disjoint variable condition replacing the nonfree hypothesis Ⅎ𝑦𝜑, not requiring ax-12 2171. (Contributed by SN, 5-Dec-2024.)
(∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
 
4-Dec-2024sucdom 9018 Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-pow 5288. (Revised by BTernaryTau, 4-Dec-2024.)
(𝐴 ∈ ω → (𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵))
 
4-Dec-2024sucdom2 8989 Strict dominance of a set over another set implies dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) Avoid ax-pow 5288. (Revised by BTernaryTau, 4-Dec-2024.)
(𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵)
 
4-Dec-2024undom 8846 Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257. (Contributed by NM, 3-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5288. (Revised by BTernaryTau, 4-Dec-2024.)
(((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼ (𝐵 ∪ 𝐷))
 
3-Dec-2024fvprc 6766 A function's value at a proper class is the empty set. See fvprcALT 6767 for a proof that uses ax-pow 5288 instead of ax-pr 5352. (Contributed by NM, 20-May-1998.) Avoid ax-pow 5288. (Revised by BTernaryTau, 3-Aug-2024.) (Proof shortened by BTernaryTau, 3-Dec-2024.)
(¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
 
3-Dec-2024f1un 6736 The union of two one-to-one functions with disjoint domains and codomains. (Contributed by BTernaryTau, 3-Dec-2024.)
(((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1→(𝐵 ∪ 𝐷))
 
3-Dec-2024dtru 5359 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 2031.

This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2709, ax-sep 5223, or ax-pow 5288. See dtruALT 5311 for a shorter proof using these axioms, and see dtruALT2 5293 for a proof that uses ax-pow 5288 instead of ax-pr 5352.

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 2171. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5352 instead of ax-pow 5288. (Revised by BTernaryTau, 3-Dec-2024.)

¬ ∀𝑥 𝑥 = 𝑦
 
2-Dec-2024onomeneq 9011 An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.) Avoid ax-pow 5288. (Revised by BTernaryTau, 2-Dec-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))
 
2-Dec-2024el 5357 Every set is an element of some other set. See elALT 5358 for a shorter proof using more axioms, and see elALT2 5292 for a proof that uses ax-9 2116 and ax-pow 5288 instead of ax-pr 5352. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Avoid ax-9 2116, ax-pow 5288. (Revised by BTernaryTau, 2-Dec-2024.)
∃𝑦 𝑥 ∈ 𝑦
 
1-Dec-2024frrlem16 9516 Lemma for general well-founded recursion. Establish a subset relationship. (Contributed by Scott Fenton, 11-Sep-2023.) Revised notion of transitive closure. (Revised by Scott Fenton, 1-Dec-2024.)
(((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ Pred (t++(𝑅 ↾ 𝐴), 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧))
 
1-Dec-2024snnen2o 9026 A singleton {𝐴} is never equinumerous with the ordinal number 2. This holds for proper singletons (𝐴 ∈ V) as well as for singletons being the empty set (𝐴 ∉ V). (Contributed by AV, 6-Aug-2019.) Avoid ax-pow 5288, ax-un 7588. (Revised by BTernaryTau, 1-Dec-2024.)
¬ {𝐴} ≈ 2o
 
1-Dec-20242onn 8472 The ordinal 2 is a natural number. For a shorter proof using Peano's postulates that depends on ax-un 7588, see 2onnALT 8473. (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7588. (Revised by BTernaryTau, 1-Dec-2024.)
2o ∈ ω
 
1-Dec-20241onn 8470 The ordinal 1 is a natural number. For a shorter proof using Peano's postulates that depends on ax-un 7588, see 1onnALT 8471. (Contributed by NM, 29-Oct-1995.) Avoid ax-un 7588. (Revised by BTernaryTau, 1-Dec-2024.)
1o ∈ ω
 
1-Dec-20242ellim 8329 A limit ordinal contains 2. (Contributed by BTernaryTau, 1-Dec-2024.)
(Lim 𝐴 → 2o ∈ 𝐴)
 
1-Dec-20241ellim 8328 A limit ordinal contains 1. (Contributed by BTernaryTau, 1-Dec-2024.)
(Lim 𝐴 → 1o ∈ 𝐴)
 
1-Dec-2024ord2eln012 8327 An ordinal that is not 0, 1, or 2 contains 2. (Contributed by BTernaryTau, 1-Dec-2024.)
(Ord 𝐴 → (2o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o ∧ 𝐴 ≠ 2o)))
 
1-Dec-2024ord1eln01 8326 An ordinal that is not 0 or 1 contains 1. (Contributed by BTernaryTau, 1-Dec-2024.)
(Ord 𝐴 → (1o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o)))
 
1-Dec-2024nlim2 8320 2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.)
¬ Lim 2o
 
1-Dec-2024nlim1 8319 1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.)
¬ Lim 1o
 
1-Dec-2024f1cdmsn 7154 If a one-to-one function with a nonempty domain has a singleton as its codomain, its domain must also be a singleton. (Contributed by BTernaryTau, 1-Dec-2024.)
((𝐹:𝐴–1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
 
30-Nov-20242on 8311 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) Avoid ax-un 7588. (Revised by BTernaryTau, 30-Nov-2024.)
2o ∈ On
 
30-Nov-20241on 8309 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.) Avoid ax-un 7588. (Revised by BTernaryTau, 30-Nov-2024.)
1o ∈ On
 
30-Nov-2024suceloni 7659 The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.) (Proof shortened by BTernaryTau, 30-Nov-2024.)
(𝐴 ∈ On → suc 𝐴 ∈ On)
 
30-Nov-2024sucexeloni 7658 If the successor of an ordinal number exists, it is an ordinal number. This variation of suceloni 7659 does not require ax-un 7588. (Contributed by BTernaryTau, 30-Nov-2024.)
((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)
 
30-Nov-2024epweon 7625 The membership relation well-orders the class of ordinal numbers. This proof does not require the axiom of regularity. Proposition 4.8(g) of [Mendelson] p. 244. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7588. (Revised by BTernaryTau, 30-Nov-2024.)
E We On
 
30-Nov-2024elex2 2818 If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) Avoid ax-9 2116, ax-ext 2709, df-clab 2716. (Revised by Wolf Lammen, 30-Nov-2024.)
(𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)
 
29-Nov-2024nndomog 8999 Cardinal ordering agrees with ordinal number ordering when the smaller number is a natural number. Compare with nndomo 9016 when both are natural numbers. (Contributed by NM, 17-Jun-1998.) Generalize from nndomo 9016. (Revised by RP, 5-Nov-2023.) Avoid ax-pow 5288. (Revised by BTernaryTau, 29-Nov-2024.)
((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵))
 
29-Nov-2024sdom0 8895 The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.) Avoid ax-pow 5288, ax-un 7588. (Revised by BTernaryTau, 29-Nov-2024.)
¬ 𝐴 ≺ ∅
 
29-Nov-20240sdomg 8891 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5288, ax-un 7588. (Revised by BTernaryTau, 29-Nov-2024.)
(𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
 
29-Nov-2024dom0 8889 A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.) Avoid ax-pow 5288, ax-un 7588. (Revised by BTernaryTau, 29-Nov-2024.)
(𝐴 ≼ ∅ ↔ 𝐴 = ∅)
 
29-Nov-20240domg 8887 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5288, ax-un 7588. (Revised by BTernaryTau, 29-Nov-2024.)
(𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)
 
29-Nov-2024en0r 8806 The empty set is equinumerous only to itself. (Contributed by BTernaryTau, 29-Nov-2024.)
(∅ ≈ 𝐴 ↔ 𝐴 = ∅)
 
29-Nov-2024brdomi 8748 Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.) Avoid ax-un 7588. (Revised by BTernaryTau, 29-Nov-2024.)
(𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
 
29-Nov-2024brdomg 8746 Dominance relation. (Contributed by NM, 15-Jun-1998.) (Proof shortened by BTernaryTau, 29-Nov-2024.)
(𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
 
29-Nov-2024brdom2g 8745 Dominance relation. This variation of brdomg 8746 does not require the Axiom of Union. (Contributed by BTernaryTau, 29-Nov-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
 
29-Nov-2024peano1 7735 Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 7735 through peano5 7740 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity. (Contributed by NM, 15-May-1994.) Avoid ax-un 7588. (Revised by BTernaryTau, 29-Nov-2024.)
∅ ∈ ω
 
28-Nov-2024phpeqd 8998 Corollary of the Pigeonhole Principle using equality. Strengthening of php 8993 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) Avoid ax-pow. (Revised by BTernaryTau, 28-Nov-2024.)
(𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐵 ⊆ 𝐴)    &   (𝜑 → 𝐴 ≈ 𝐵)    ⇒   (𝜑 → 𝐴 = 𝐵)
 
27-Nov-2024frmin 9507 Every (possibly proper) subclass of a class 𝐴 with a well-founded set-like relation 𝑅 has a minimal element. This is a very strong generalization of tz6.26 6250 and tz7.5 6287. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 27-Nov-2024.)
(((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
 
26-Nov-2024php3 8995 Corollary of Pigeonhole Principle. If 𝐴 is finite and 𝐵 is a proper subset of 𝐴, the 𝐵 is strictly less numerous than 𝐴. Stronger version of Corollary 6C of [Enderton] p. 135. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5288. (Revised by BTernaryTau, 26-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)
 
25-Nov-2024domsdomtrfi 8988 Transitivity of dominance and strict dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domsdomtr 8899). (Contributed by BTernaryTau, 25-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶)
 
25-Nov-2024sdomdomtrfi 8987 Transitivity of strict dominance and dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike sdomdomtr 8897). (Contributed by BTernaryTau, 25-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐶)
 
25-Nov-2024predres 6242 Predecessor class is unaffected by restriction to the base class. (Contributed by Scott Fenton, 25-Nov-2024.)
Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋)
 
25-Nov-2024predprc 6241 The predecessor of a proper class is empty. (Contributed by Scott Fenton, 25-Nov-2024.)
(¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)
 
25-Nov-2024predrelss 6240 Subset carries from relation to predecessor class. (Contributed by Scott Fenton, 25-Nov-2024.)
(𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋))
 
24-Nov-2024ssdomfi2 8983 A set dominates its finite subsets, proved without using the Axiom of Power Sets (unlike ssdomg 8786). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ≼ 𝐵)
 
24-Nov-2024domtrfir 8980 Transitivity of dominance relation for finite sets, proved without using the Axiom of Power Sets (unlike domtr 8793). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐶 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
 
24-Nov-2024domtrfi 8979 Transitivity of dominance relation when 𝐵 is finite, proved without using the Axiom of Power Sets (unlike domtr 8793). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
 
24-Nov-2024domtrfil 8978 Transitivity of dominance relation when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domtr 8793). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
 
24-Nov-2024f1domfi2 8968 If the domain of a one-to-one function is finite, then the function's domain is dominated by its codomain when the latter is a set. This theorem is proved without using the Axiom of Power Sets (unlike f1dom2g 8757). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
 
24-Nov-2024rabid2 3314 An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2024.)
(𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
 
24-Nov-2024clelsb2 2867 Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2123). (Contributed by Jim Kingdon, 22-Nov-2018.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 24-Nov-2024.)
([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)
 
23-Nov-2024natglobalincr 46512 Local monotonicity on half-open integer range implies global monotonicity. (Contributed by Ender Ting, 23-Nov-2024.)
∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘) < (𝐵‘(𝑘 + 1))    &   𝑇 ∈ ℤ    ⇒   ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ ((𝑘 + 1)...𝑇)(𝐵‘𝑘) < (𝐵‘𝑡)
 
23-Nov-2024prjcrv0 40470 The "curve" (zero set) corresponding to the zero polynomial contains all coordinates. (Contributed by SN, 23-Nov-2024.)
𝑌 = ((0...𝑁) mPoly 𝐾)    &    0 = (0g‘𝑌)    &   𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ Field)    ⇒   (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = 𝑃)
 
23-Nov-2024prjcrvval 40469 Value of the projective curve function. (Contributed by SN, 23-Nov-2024.)
𝐻 = ((0...𝑁) mHomP 𝐾)    &   𝐸 = ((0...𝑁) eval 𝐾)    &   𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &    0 = (0g‘𝐾)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝐹 ∈ ∪ ran 𝐻)    ⇒   (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }})
 
23-Nov-2024prjcrvfval 40468 Value of the projective curve function. (Contributed by SN, 23-Nov-2024.)
𝐻 = ((0...𝑁) mHomP 𝐾)    &   𝐸 = ((0...𝑁) eval 𝐾)    &   𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &    0 = (0g‘𝐾)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ Field)    ⇒   (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))
 
23-Nov-2024df-prjcrv 40467 Define the projective curve function. This takes a homogeneous polynomial and outputs the homogeneous coordinates where the polynomial evaluates to zero (the "zero set"). (In other words, scalar multiples are collapsed into the same projective point. See mhphf4 40288 and prjspvs 40449). (Contributed by SN, 23-Nov-2024.)
ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ0, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}))
 
23-Nov-2024mhphf4 40288 A homogeneous polynomial defines a homogeneous function; this is mhphf3 40287 with evalSub collapsed to eval. (Contributed by SN, 23-Nov-2024.)
𝑄 = (𝐼 eval 𝑆)    &   𝐻 = (𝐼 mHomP 𝑆)    &   𝐾 = (Base‘𝑆)    &   𝐹 = (𝑆 freeLMod 𝐼)    &   𝑀 = (Base‘𝐹)    &    ∙ = ( ·𝑠 ‘𝐹)    &    · = (.r‘𝑆)    &    ↑ = (.g‘(mulGrp‘𝑆))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝐿 ∈ 𝐾)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   (𝜑 → 𝐴 ∈ 𝑀)    ⇒   (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
 
23-Nov-2024mhphf3 40287 A homogeneous polynomial defines a homogeneous function; this is mhphf2 40286 with the finite support restriction (frlmpws 20957, frlmbas 20962) on the assignments 𝐴 from variables to values. See comment of mhphf2 40286. (Contributed by SN, 23-Nov-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝐻 = (𝐼 mHomP 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝐹 = (𝑆 freeLMod 𝐼)    &   𝑀 = (Base‘𝐹)    &    ∙ = ( ·𝑠 ‘𝐹)    &    · = (.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 ∈ 𝑀)    ⇒   (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
 
23-Nov-2024evl0 40271 The zero polynomial evaluates to zero. (Contributed by SN, 23-Nov-2024.)
𝑄 = (𝐼 eval 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑅)    &   𝑂 = (0g‘𝑅)    &    0 = (0g‘𝑊)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → (𝑄‘ 0 ) = ((𝐵 ↑m 𝐼) × {𝑂}))
 
23-Nov-2024mplascl0 40270 The zero scalar as a polynomial. (Contributed by SN, 23-Nov-2024.)
𝑊 = (𝐼 mPoly 𝑅)    &   𝐴 = (algSc‘𝑊)    &   𝑂 = (0g‘𝑅)    &    0 = (0g‘𝑊)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → (𝐴‘𝑂) = 0 )
 
23-Nov-2024fldcrngd 40255 A field is a commutative ring. EDITORIAL: Shortens recrng 20826. Also recrng 20826 should be named resrng. Also fldcrng 36162 is misnamed. (Contributed by SN, 23-Nov-2024.)
(𝜑 → 𝑅 ∈ Field)    ⇒   (𝜑 → 𝑅 ∈ CRing)
 
23-Nov-2024sn-iotaval 40195 iotaval 6407 without ax-10 2137, ax-11 2154, ax-12 2171. (Contributed by SN, 23-Nov-2024.)
(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
 
23-Nov-2024abbi1sn 40191 Originally part of uniabio 6406. Convert a theorem about df-iota 6391 to one about dfiota2 6392, without ax-10 2137, ax-11 2154, ax-12 2171. Although, eu6 2574 uses ax-10 2137 and ax-12 2171. (Contributed by SN, 23-Nov-2024.)
(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})
 
23-Nov-2024recvs 24309 The field of the real numbers as left module over itself is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.) (Proof shortened by SN, 23-Nov-2024.)
𝑅 = (ringLMod‘ℝfld)    ⇒   𝑅 ∈ ℂVec
 
23-Nov-2024moel 3358 "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2154. (Revised by Wolf Lammen, 23-Nov-2024.)
(∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)
 
23-Nov-2024rmobidva 3327 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.) Avoid ax-6 1971, ax-7 2011, ax-12 2171. (Revised by Wolf Lammen, 23-Nov-2024.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
 
23-Nov-2024nfrabw 3318 A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3320 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Nov-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    ⇒   Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
 
22-Nov-2024tworepnotupword 46521 Word of two matching characters is never an increasing sequence. (Contributed by Ender Ting, 22-Nov-2024.)
𝐴 ∈ V    ⇒    ¬ (⟨“𝐴”⟩ ++ ⟨“𝐴”⟩) ∈ UpWord𝑆
 
22-Nov-2024singoutnupword 46518 Singleton with character out of range 𝑆 is not an increasing sequence for that range. (Contributed by Ender Ting, 22-Nov-2024.)
𝐴 ∈ V    ⇒   (¬ 𝐴 ∈ 𝑆 → ¬ ⟨“𝐴”⟩ ∈ UpWord𝑆)
 
22-Nov-2024natlocalincr 46511 Global monotonicity on half-open range implies local monotonicity. (Contributed by Ender Ting, 22-Nov-2024.)
∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ (1..^(𝑇 + 1))(𝑘 < 𝑡 → (𝐵‘𝑘) < (𝐵‘𝑡))    ⇒   ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘) < (𝐵‘(𝑘 + 1))
 
22-Nov-2024et-ltneverrefl 46510 Less-than class is never reflexive. (Contributed by Ender Ting, 22-Nov-2024.) Prefer to specify theorem domain and then apply ltnri 11084. (New usage is discouraged.)
¬ 𝐴 < 𝐴
 
22-Nov-2024domnsymfi 8986 If a set dominates a finite set, it cannot also be strictly dominated by the finite set. This theorem is proved without using the Axiom of Power Sets (unlike domnsym 8886). (Contributed by BTernaryTau, 22-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ≺ 𝐴)
 
21-Nov-2024upwordsseti 46520 Strictly increasing sequences with a set given for range form a set. (Contributed by Ender Ting, 21-Nov-2024.)
𝑆 ∈ V    ⇒   UpWord𝑆 ∈ V
 
21-Nov-2024upwordsing 46519 Singleton is an increasing sequence for any compatible range. (Contributed by Ender Ting, 21-Nov-2024.)
𝐴 ∈ 𝑆    ⇒   ⟨“𝐴”⟩ ∈ UpWord𝑆
 
21-Nov-2024singoutnword 46517 Singleton with character out of range 𝑉 is not a word for that range. (Contributed by Ender Ting, 21-Nov-2024.)
𝐴 ∈ V    ⇒   (¬ 𝐴 ∈ 𝑉 → ¬ ⟨“𝐴”⟩ ∈ Word 𝑉)
 
21-Nov-2024nfreuw 3305 Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3308 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2116, ax-ext 2709. (Revised by Wolf Lammen, 21-Nov-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑
 
21-Nov-2024nfrmow 3304 Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3309 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2116, ax-ext 2709. (Revised by Wolf Lammen, 21-Nov-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
 
21-Nov-2024eeor 2330 Distribute existential quantifiers. (Contributed by NM, 8-Aug-1994.) Avoid ax-10 2137. (Revised by Gino Giotto, 21-Nov-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    ⇒   (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
 
21-Nov-2024aaan 2328 Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993.) Avoid ax-10 2137. (Revised by Gino Giotto, 21-Nov-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    ⇒   (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
 
20-Nov-2024php2 8994 Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.) Avoid ax-pow 5288. (Revised by BTernaryTau, 20-Nov-2024.)
((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)
 
20-Nov-20242ralor 3296 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))
 
20-Nov-2024sbrim 2301 Substitution in an implication with a variable not free in the antecedent affects only the consequent. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) Avoid ax-10 2137. (Revised by Gino Giotto, 20-Nov-2024.)
Ⅎ𝑥𝜑    ⇒   ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 
19-Nov-2024upwordisword 46516 Any increasing sequence is a sequence. (Contributed by Ender Ting, 19-Nov-2024.)
(𝐴 ∈ UpWord𝑆 → 𝐴 ∈ Word 𝑆)
 
19-Nov-2024upwordnul 46515 Empty set is an increasing sequence for every range. (Contributed by Ender Ting, 19-Nov-2024.)
∅ ∈ UpWord𝑆
 
19-Nov-2024df-upword 46514 Strictly increasing sequence is a sequence, adjacent elements of which increase. (Contributed by Ender Ting, 19-Nov-2024.)
UpWord𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))}
 
19-Nov-2024fri 5549 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-2024php 8993 Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of phplem1 8990, phplem2 8991, nneneq 8992, and this final piece of the proof. (Contributed by NM, 29-May-1998.) Avoid ax-pow 5288. (Revised by BTernaryTau, 18-Nov-2024.)
((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐴 ≈ 𝐵)
 
18-Nov-2024wfr3 8168 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 8166 and wfr2 8167 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 8166 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 8165 A weak version of wfr2 8167 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 8164 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 8137 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 8126 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 8125 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 8123 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 8122 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 8120 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 8118 Weak version of fpr2 8120 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 8100 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 2137. (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 1923 and exbidv 1924 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2172. (Revised by Wolf Lammen, 30-Mar-2024.) Avoid ax-10 2137. (Revised by Gino Giotto, 18-Nov-2024.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
 
18-Nov-2024equsexv 2260 An equivalence related to implicit substitution. Version of equsex 2418 with a disjoint variable condition, which does not require ax-13 2372. See equsexvw 2008 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2259. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) Avoid ax-10 2137. (Revised by Gino Giotto, 18-Nov-2024.)
Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
 
17-Nov-2024bj-rdg0gALT 35242 Alternate proof of rdg0g 8258. More direct since it bypasses tz7.44-1 8237 and rdg0 8252 (and vtoclg 3505, vtoclga 3513). (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-2024wfrfun 8163 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 8162 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 8161 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 8160 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 8132 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 8130 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 8124 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 6259 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 6256 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 6253 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 6250 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 5187 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 5189 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 5153 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 5548 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 5547 Alternate definition of df-fr 5544. See dffr5 33721 for a definition without dummy variables (but note that their equivalence uses ax-sep 5223). (Contributed by BJ, 16-Nov-2024.)
(𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
 
15-Nov-20241strbas 16929 The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Proof shortened by AV, 15-Nov-2024.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
 
15-Nov-20241strstr1 16928 A constructed one-slot structure. (Contributed by AV, 15-Nov-2024.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   𝐺 Struct ⟨(Base‘ndx), (Base‘ndx)⟩
 
14-Nov-2024aks4d1 40097 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 40096 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 40070 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 40094 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 40093 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-2024prstcocval 46352 Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ⊥ = (oc‘𝐾))    ⇒   (𝜑 → ⊥ = (oc‘𝐶))
 
12-Nov-2024prstcleval 46349 Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐾))    ⇒   (𝜑 → ≤ = (le‘𝐶))
 
12-Nov-2024zlmtset 31914 Topology in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof shortened by AV, 12-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐽 = (TopSet‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → 𝐽 = (TopSet‘𝑊))
 
12-Nov-2024setsmsbas 23628 The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof shortened by AV, 12-Nov-2024.)
(𝜑 → 𝑋 = (Base‘𝑀))    &   (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    ⇒   (𝜑 → 𝑋 = (Base‘𝐾))
 
12-Nov-2024matvsca 21564 The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    ⇒   ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐴))
 
12-Nov-2024matsca 21562 The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    ⇒   ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴))
 
12-Nov-2024sravsca 20449 The scalar product 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.) (Proof shortened by AV, 12-Nov-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
 
12-Nov-2024srasca 20447 The set of scalars 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.) (Proof shortened by AV, 12-Nov-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
 
12-Nov-2024odubas 18009 Base set of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Proof shortened by AV, 12-Nov-2024.)
𝐷 = (ODual‘𝑂)    &   𝐵 = (Base‘𝑂)    ⇒   𝐵 = (Base‘𝐷)
 
12-Nov-2024slotsdifocndx 17128 The index of the slot for the orthocomplementation is not the index of other slots. Formerly part of proof for prstcocval 46352. (Contributed by AV, 12-Nov-2024.)
((oc‘ndx) ≠ (comp‘ndx) ∧ (oc‘ndx) ≠ (Hom ‘ndx))
 
12-Nov-2024slotsdifplendx2 17127 The index of the slot for the "less than or equal to" ordering is not the index of other slots. Formerly part of proof for prstcleval 46349. (Contributed by AV, 12-Nov-2024.)
((le‘ndx) ≠ (comp‘ndx) ∧ (le‘ndx) ≠ (Hom ‘ndx))
 
12-Nov-2024slotsdifipndx 17045 The slot for the scalar is not the index of other slots. Formerly part of proof for srasca 20447 and sravsca 20449. (Contributed by AV, 12-Nov-2024.)
(( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx))
 
12-Nov-2024ssdomfi 8982 A finite set dominates its subsets, proved without using the Axiom of Power Sets (unlike ssdomg 8786). (Contributed by BTernaryTau, 12-Nov-2024.)
(𝐵 ∈ Fin → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵))
 
11-Nov-2024mpteq1df 42779 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 40286 A homogeneous polynomial defines a homogeneous function; this is mhphf 40285 with simpler notation in the conclusion in exchange for a complex definition of ∙, which is based on frlmvscafval 20973 but without the finite support restriction (frlmpws 20957, frlmbas 20962) on the assignments 𝐴 from variables to values.

TODO?: Polynomials (df-mpl 21114) 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-2024zlmds 31912 Distance in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof shortened by AV, 11-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐷 = (dist‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊))
 
11-Nov-2024setsmsds 23630 The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof shortened by AV, 11-Nov-2024.)
(𝜑 → 𝑋 = (Base‘𝑀))    &   (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    ⇒   (𝜑 → (dist‘𝑀) = (dist‘𝐾))
 
11-Nov-2024thlle 20903 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)    &   𝐼 = (toInc‘𝐶)    &    ≤ = (le‘𝐼)    ⇒    ≤ = (le‘𝐾)
 
11-Nov-2024thlbas 20901 Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)    ⇒   𝐶 = (Base‘𝐾)
 
11-Nov-2024cnfldfunALT 20610 The field of complex numbers is a function. Alternate proof of cnfldfun 20609 not requiring that the index set of the components is ordered, but using quadratically many inequalities for the indices. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 11-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Fun ℂfld
 
11-Nov-2024fldidom 20576 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-2024slotsdifdsndx 17104 The index of the slot for the distance is not the index of other slots. Formerly part of proof for cnfldfunALT 20610. (Contributed by AV, 11-Nov-2024.)
((*𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx))
 
11-Nov-2024plendxnocndx 17094 The slot for the orthocomplementation is not the slot for the order in an extensible structure. Formerly part of proof for thlle 20903. (Contributed by AV, 11-Nov-2024.)
(le‘ndx) ≠ (oc‘ndx)
 
11-Nov-2024basendxnocndx 17093 The slot for the orthocomplementation is not the slot for the base set in an extensible structure. Formerly part of proof for thlbas 20901. (Contributed by AV, 11-Nov-2024.)
(Base‘ndx) ≠ (oc‘ndx)
 
11-Nov-2024slotsdifplendx 17085 The index of the slot for the distance is not the index of other slots. Formerly part of proof for cnfldfunALT 20610. (Contributed by AV, 11-Nov-2024.)
((*𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx))
 
11-Nov-2024tsetndxnstarvndx 17069 The slot for the topology is not the slot for the involution in an extensible structure. Formerly part of proof for cnfldfunALT 20610. (Contributed by AV, 11-Nov-2024.)
(TopSet‘ndx) ≠ (*𝑟‘ndx)
 
11-Nov-2024nneneq 8992 Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.) Avoid ax-pow 5288. (Revised by BTernaryTau, 11-Nov-2024.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))
 
11-Nov-2024ofeqd 7535 Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
(𝜑 → 𝑅 = 𝑆)    ⇒   (𝜑 → ∘f 𝑅 = ∘f 𝑆)
 
11-Nov-2024dffun2 6443 Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 11-Nov-2024.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
 
11-Nov-2024iunopab 5472 Move indexed union inside an ordered-pair class abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.) Avoid ax-sep 5223, ax-nul 5230, ax-pr 5352. (Revised by SN, 11-Nov-2024.)
∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑}
 
11-Nov-2024mpteq2ia 5177 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 5174 Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.) Remove dependency on ax-10 2137. (Revised by SN, 11-Nov-2024.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
 
11-Nov-2024mpteq2da 5172 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 5170 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 5167 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 5163 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) Remove dependency on ax-10 2137, ax-12 2171. (Revised by SN, 11-Nov-2024.)
(𝜑 → 𝐴 = 𝐶)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
 
11-Nov-2024mpteq12df 5160 An equality inference for the maps-to notation. Compare mpteq12dv 5165. (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 5159 An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) Remove dependency on ax-10 2137. (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 40095 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 40092 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)))
 
10-Nov-2024slotsdifunifndx 17111 The index of the slot for the uniform set is not the index of other slots. Formerly part of proof for cnfldfunALT 20610. (Contributed by AV, 10-Nov-2024.)
(((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)))
 
9-Nov-2024bj-flddrng 35460 Fields are division rings (elemental version). (Contributed by BJ, 9-Nov-2024.)
(𝐹 ∈ Field → 𝐹 ∈ DivRing)
 
9-Nov-2024bj-dfid2ALT 35236 Alternate version of dfid2 5491. (Contributed by BJ, 9-Nov-2024.) (Proof modification is discouraged.) Use df-id 5489 instead to make the semantics of the construction df-opab 5137 clearer. (New usage is discouraged.)
I = {⟨𝑥, 𝑥⟩ ∣ ⊤}
 
9-Nov-2024ttgval 27236 Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof shortened by AV, 9-Nov-2024.)
𝐺 = (toTG‘𝐻)    &   𝐵 = (Base‘𝐻)    &    − = (-g‘𝐻)    &    · = ( ·𝑠 ‘𝐻)    &   𝐼 = (Itv‘𝐺)    ⇒   (𝐻 ∈ 𝑉 → (𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) ∧ 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})))
 
9-Nov-2024lngndxnitvndx 26804 The slot for the line is not the slot for the Interval (segment) in an extensible structure. Formerly part of proof for ttgval 27236. (Contributed by AV, 9-Nov-2024.)
(LineG‘ndx) ≠ (Itv‘ndx)
 
9-Nov-2024rescabs 17547 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 9-Nov-2024.)
(𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   (𝜑 → 𝐽 Fn (𝑇 × 𝑇))    &   (𝜑 → 𝑆 ∈ 𝑊)    &   (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))
 
7-Nov-2024ressbas 16947 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 16910 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 40197 iotaex 6413 without ax-10 2137, ax-11 2154, ax-12 2171. (Contributed by SN, 6-Nov-2024.)
(℩𝑥𝜑) ∈ V
 
6-Nov-2024sn-iotassuni 40196 iotassuni 6412 without ax-10 2137, ax-11 2154, ax-12 2171. (Contributed by SN, 6-Nov-2024.)
(℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}
 
6-Nov-2024sn-iotanul 40194 Version of iotanul 6411 using df-iota 6391 instead of dfiota2 6392. (Contributed by SN, 6-Nov-2024.)
(¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
 
6-Nov-2024sn-iotauni 40193 Version of iotauni 6408 using df-iota 6391 instead of dfiota2 6392. (Contributed by SN, 6-Nov-2024.)
(∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
 
6-Nov-2024iotavallem 40192 Version of iotaval 6407 using df-iota 6391 instead of dfiota2 6392. (Contributed by SN, 6-Nov-2024.)
({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
 
6-Nov-2024sn-iotalemcor 40190 Corollary of sn-iotalem 40189. Compare sb8iota 6403. (Contributed by SN, 6-Nov-2024.)
(℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦})
 
6-Nov-2024sn-iotalem 40189 An unused lemma showing that many equivalences involving df-iota 6391 are potentially provable without ax-10 2137, ax-11 2154, ax-12 2171. (Contributed by SN, 6-Nov-2024.)
{𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
 
6-Nov-2024eqimssd 40183 Equality implies inclusion, deduction version. (Contributed by SN, 6-Nov-2024.)
(𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → 𝐴 ⊆ 𝐵)
 
6-Nov-2024hlhilsmul 39958 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 39956 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 39954 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 39952 Lemma for hlhilsbase 39954 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 19871 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 19869 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 19867 Lemma for opprbas 19869 and oppradd 19871. (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 19004 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 18493 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 5491 Alternate definition of the identity relation. Instance of dfid3 5492 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 5489 instead to make the semantics of the constructor df-opab 5137 clearer. (New usage is discouraged.)

I = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥}
 
5-Nov-2024r19.30 3268 Restricted quantifier version of 19.30 1884. (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-2024phplem2 8991 Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) Avoid ax-pow 5288. (Revised by BTernaryTau, 4-Nov-2024.)
𝐴 ∈ V    ⇒   ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))
 
4-Nov-2024sbthfi 8985 Schroeder-Bernstein Theorem for finite sets, proved without using the Axiom of Power Sets (unlike sbth 8880). (Contributed by BTernaryTau, 4-Nov-2024.)
((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)
 
4-Nov-2024sbthfilem 8984 Lemma for sbthfi 8985. (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 3266 A commonly used pattern based on r19.29 3184, 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 3264 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 3257 Restricted quantifier version of 19.12 2321. (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 3255 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 3200 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 2054 Existential quantification over a given variable is idempotent. Weak version of bj-exexbiex 34882, requiring fewer axioms. (Contributed by Gino Giotto, 4-Nov-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥𝜑 ↔ ∃𝑥∃𝑥𝜑)
 
3-Nov-2024znmul 20748 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 20746 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 20744 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 20742 Lemma for znbas 20751. (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 20724 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 20722 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 20720 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 20718 Lemma for zlmbas 20720 and zlmplusg 20722. (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 3195 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 3173 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 21159 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 20726 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 3631 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 3628 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 41842 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 41840 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 41835 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 41829 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 41827 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 21260 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 21258 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 21256 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 21254 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 21252 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 21250 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 17084 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 21258. (Contributed by AV, 1-Nov-2024.)
(le‘ndx) ≠ ( ·𝑠 ‘ndx)
 
1-Nov-2024plendxnscandx 17083 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 21260. (Contributed by AV, 1-Nov-2024.)
(le‘ndx) ≠ (Scalar‘ndx)
 
1-Nov-2024plendxnmulrndx 17082 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 21256. (Contributed by AV, 1-Nov-2024.)
(le‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024mendvscafval 41015 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 41014 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 41012 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 41010 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 40091 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 40090 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-2024resvmulr 31538 .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 31536 ·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 31534 +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 31532 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 31530 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 23854 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 23809 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 23807 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 23805 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 23803 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 23802 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 23800 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 23798 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 23796 Lemma for tngbas 23798 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 22163 The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 22161 from the direct component assignment version indistps2 22162. (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 22092 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 17101 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 17068 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 17066 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 17065 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 17064 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 16893 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 31530. (Contributed by AV, 31-Oct-2024.)
(𝐸‘∅) = ∅    &   𝑍 = (𝑋𝑂𝑌)    &   Rel dom 𝑂    ⇒   (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑍))
 
31-Oct-2024fveqprc 16892 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 20718. (Contributed by AV, 31-Oct-2024.)
(𝐸‘∅) = ∅    &   𝑌 = (𝐹‘𝑋)    ⇒   (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑌))
 
31-Oct-2024ttrclse 9485 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 9484 Lemma for ttrclse 9485. 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 9483 Lemma for ttrclse 9485. 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 8265 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-2024reximia 3176 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 3164 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 3159 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 3154 Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 42480. Version of nfra2 3157 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 40089 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 40088 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-2024basendxltedgfndx 27363 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 18043 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 17080 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 17079 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 17078 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 3026 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 27254 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 27247 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 27245 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 27242 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 27240 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 27238 Lemma for ttgbas 27240, ttgvsca 27245 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 26803 The slots Base, +g, ·𝑠 and dist are different from the slot LineG. Formerly part of ttglem 27238 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 26802 The slots Base, +g, ·𝑠 and dist are different from the slot Itv. Formerly part of ttglem 27238 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 23811 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 20455 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 20452 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 20445 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 20443 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 20441 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 20439 Lemma for srabase 20441 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 17103 The slot for the distance function is not the slot for the topology in an extensible structure. Formerly part of proof for tngds 23811. (Contributed by AV, 29-Oct-2024.)
(dist‘ndx) ≠ (TopSet‘ndx)
 
29-Oct-2024slotsdnscsi 17102 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot dist. Formerly part of sralem 20439 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 17070 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. Formerly part of sralem 20439 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 17044 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 19728. (Contributed by AV, 29-Oct-2024.)
(·𝑖‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024ipndxnplusgndx 17043 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 17033 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 20191. (Contributed by AV, 29-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024scandxnmulrndx 17028 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 19728. (Contributed by AV, 29-Oct-2024.)
(Scalar‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024pm13.18 3025 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 40087 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 27361 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 23418 Lemma for tusbas 23420, tusunif 23421, and tustopn 23423. (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 17853 Lemma 1 for estrres 17856. (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 17125 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 17110 The slot for the uniform set is not the slot for the topology in an extensible structure. Formerly part of proof for tuslem 23418. (Contributed by AV, 28-Oct-2024.)
(UnifSet‘ndx) ≠ (TopSet‘ndx)
 
28-Oct-2024basendxltunifndx 17108 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 23418. (Contributed by AV, 28-Oct-2024.)
(Base‘ndx) < (UnifSet‘ndx)
 
28-Oct-2024unifndxnn 17107 The index of the slot for the uniform set in an extensible structure is a positive integer. Formerly part of proof for tuslem 23418. (Contributed by AV, 28-Oct-2024.)
(UnifSet‘ndx) ∈ ℕ
 
28-Oct-2024dsndxnbasendx 17099 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 17098 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 23637. (Contributed by AV, 28-Oct-2024.)
(Base‘ndx) < (dist‘ndx)
 
28-Oct-2024dsndxnn 17097 The index of the slot for the distance in an extensible structure is a positive integer. Formerly part of proof for tmslem 23637. (Contributed by AV, 28-Oct-2024.)
(dist‘ndx) ∈ ℕ
 
28-Oct-2024basendxnmulrndx 17005 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 16960 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 6226 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 6225 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 2968 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 2941 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 40086 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 40085 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 16998 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 16996 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 16995 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 8128 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 8166, wfr2 8167, and wfr3 8168. (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 7965 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 7964 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 6220 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 40124 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 9482 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 9481 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 9480 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 9479 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 9468 Bound variable hypothesis builder for transitive closure. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.)
(𝜑 → Ⅎ𝑥𝑅)    ⇒   (𝜑 → Ⅎ𝑥t++𝑅)
 
26-Oct-2024nfopab 5143 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 5142 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 3782 Class version of one implication of equvelv 2034. (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 3772 Distribution of class substitution over implication. One direction of sbcimg 3767 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) Avoid ax-10 2137, ax-12 2171. (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 2812 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 2324 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 40123 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 40122 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 8494 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 40121 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 40120 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 40119 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 17109 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 17042 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 17026 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 40118 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 9478 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 9477 Composition law for the transitive closure of a relationship. (Contributed by Scott Fenton, 20-Oct-2024.)
(𝑅 ∘ t++𝑅) ⊆ t++𝑅
 
20-Oct-2024ttrclco 9476 Composition law for the transitive closure of a relationship. (Contributed by Scott Fenton, 20-Oct-2024.)
(t++𝑅 ∘ 𝑅) ⊆ t++𝑅
 
20-Oct-2024ttrclresv 9475 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 16951 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 20191 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 20125 of a left module, see also islmod 20127. (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 19735 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 18954 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 17541 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 17428 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 17100 The slot for the distance function is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpds 19733. (Contributed by AV, 18-Oct-2024.)
(dist‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024plendxnplusgndx 17081 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 31238. (Contributed by AV, 18-Oct-2024.)
(le‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024tsetndxnplusgndx 17067 The slot for the topology is not the slot for the group operation in an extensible structure. Formerly part of proof for oppgtset 18958. (Contributed by AV, 18-Oct-2024.)
(TopSet‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024vscandxnscandx 17034 The slot for the scalar product is not the slot for the scalar field in an extensible structure. Formerly part of proof for rmodislmod 20191. (Contributed by AV, 18-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (Scalar‘ndx)
 
18-Oct-2024vscandxnplusgndx 17032 The slot for the scalar product is not the slot for the group operation in an extensible structure. Formerly part of proof for rmodislmod 20191. (Contributed by AV, 18-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024vscandxnbasendx 17031 The slot for the scalar product is not the slot for the base set in an extensible structure. Formerly part of proof for rmodislmod 20191. (Contributed by AV, 18-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (Base‘ndx)
 
18-Oct-2024scandxnplusgndx 17027 The slot for the scalar field is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpsca 19728. (Contributed by AV, 18-Oct-2024.)
(Scalar‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024starvndxnmulrndx 17016 The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17018. (Contributed by AV, 18-Oct-2024.)
(*𝑟‘ndx) ≠ (.r‘ndx)
 
18-Oct-2024starvndxnplusgndx 17015 The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17018. (Contributed by AV, 18-Oct-2024.)
(*𝑟‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024starvndxnbasendx 17014 The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17018. (Contributed by AV, 18-Oct-2024.)
(*𝑟‘ndx) ≠ (Base‘ndx)
 
17-Oct-2024nnuni 33692 The union of a finite ordinal is a finite ordinal. (Contributed by Scott Fenton, 17-Oct-2024.)
(𝐴 ∈ ω → ∪ 𝐴 ∈ ω)
 
17-Oct-2024basendxnplusgndx 16992 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 16991 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 16990 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 16956 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 16937 A constructed two-slot structure. Version of 2strstr 16934 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 16932 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 16924 The index of the base set is an element in a weak universe containing the natural numbers. Formerly part of proof for 1strwun 16932. (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 16900 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)⟩))
 
17-Oct-2024ttrcltr 9474 The transitive closure of a class is transitive. (Contributed by Scott Fenton, 17-Oct-2024.)
(t++𝑅 ∘ t++𝑅) ⊆ t++𝑅
 
17-Oct-2024ssttrcl 9473 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 9470 The transitive closure of a class is a relation. (Contributed by Scott Fenton, 17-Oct-2024.)
Rel t++𝑅
 
17-Oct-2024nfttrcl 9469 Bound variable hypothesis builder for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
Ⅎ𝑥𝑅    ⇒   Ⅎ𝑥t++𝑅
 
17-Oct-2024ttrcleq 9467 Equality theorem for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
(𝑅 = 𝑆 → t++𝑅 = t++𝑆)
 
17-Oct-2024df-ttrcl 9466 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 8493 There is at most one left additive inverse for natural number addition. (Contributed by Scott Fenton, 17-Oct-2024.)
(𝐴 ∈ ω → ∃*𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
 
16-Oct-2024thincciso 46330 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 35113 Alternate proof of elabd2 3601 bypassing elab6g 3600 (and using sbiedvw 2096 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 7733 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 7635 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 6219 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 6218 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 6217 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 6216 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 6215 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 6003 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 6001 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 5997 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 5995, remove, and relabel elimasn1 5995 to "elimasn".
𝐵 ∈ V    &   𝐶 ∈ V    ⇒   (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
 
16-Oct-2024elimasn1 5995 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Use df-br 5075 and shorten. (Revised by BJ, 16-Oct-2024.)
𝐵 ∈ V    &   𝐶 ∈ V    ⇒   (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
 
16-Oct-2024elimasng1 5994 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) Revise to use df-br 5075 and to prove elimasn1 5995 from it. (Revised by BJ, 16-Oct-2024.)
((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
 
16-Oct-2024elabd2 3601 Membership in a class abstraction, using implicit substitution. Deduction version of elab 3609. (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 7382 The law of concretion for operation class abstraction. Compare elopab 5440. (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 2137, ax-11 2154. (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 5147 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 4237 A class abstraction defined by a negation. Version of notab 4238 using implicit substitution, which does not require ax-10 2137, ax-12 2171. (Contributed by Gino Giotto, 15-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜓})
 
15-Oct-2024unabw 4231 Union of two class abstractions. Version of unab 4232 using implicit substitution, which does not require ax-8 2108, ax-10 2137, ax-12 2171. (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 3870 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 3868 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 3851 Substitution doesn't affect a constant 𝐵 (in which 𝑥 does not occur). csbconstgf 3850 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.) Avoid ax-12 2171. (Revised by Gino Giotto, 15-Oct-2024.)
(𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
 
14-Oct-2024catcxpccl 17924 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 17834 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 17831 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 17832. (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 17830 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 17832. (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 17829 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 17832. (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 17828 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 17832. (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 17827 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 17832. (Contributed by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝑋 ∈ 𝑈)
 
14-Oct-2024fuchom 17678 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 17423 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 27360 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 27359 Index value of the df-edgf 27357 slot. (Contributed by AV, 13-Oct-2024.) (New usage is discouraged.)
(.ef‘ndx) = 18
 
13-Oct-20240pos 18039 Technical lemma to simplify the statement of ipopos 18254. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 16890) 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 17832 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 17672 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 17614 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 17545 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 17165 Lemma for prdsval 17166. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 17166, dependency on df-hom 16986 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 16922 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 16561 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 6234 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 3799 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 3795 Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3793. (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 3790 Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1813). (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 3761 Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) Avoid ax-10 2137, ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
 
12-Oct-2024sbcieg 3756 Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2137, ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
 
12-Oct-2024elabgt 3603 Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3607.) (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 3602 Membership in a class abstraction, using implicit substitution. Deduction version of elab 3609. (Contributed by Gino Giotto, 12-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒))
 
12-Oct-2024ceqsexv 3479 Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) Avoid ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
 
12-Oct-2024rabrabi 3427 Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2137, ax-11 2154 and ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.)
(𝑥 = 𝑦 → (𝜒 ↔ 𝜑))    ⇒   {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)}
 
12-Oct-2024sbco4lem 2273 Lemma for sbco4 2275. 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 40113 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 41920 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 41919 Express the predicate on 𝑈 and 𝑧 in ismnu 41879 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 34115 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 34077 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 34074 No surreal is a member of its right set. (Contributed by Scott Fenton, 9-Oct-2024.)
¬ 𝑋 ∈ ( R ‘𝑋)
 
9-Oct-2024leftirr 34073 No surreal is a member of its left set. (Contributed by Scott Fenton, 9-Oct-2024.)
¬ 𝑋 ∈ ( L ‘𝑋)
 
9-Oct-2024madeun 34066 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 34064 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 34063 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 34062 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 34061 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 34060 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 34059 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 34048 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 34047 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 34046 New sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( N ‘𝐴) ⊆ No
 
9-Oct-2024oldssno 34045 Old sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( O ‘𝐴) ⊆ No
 
9-Oct-2024madessno 34044 Made sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( M ‘𝐴) ⊆ No
 
9-Oct-2024newval 34039 The value of the new options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))
 
9-Oct-2024dtruALT2 5293 Alternate proof of dtru 5359 using ax-pow 5288 instead of ax-pr 5352. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2372. (Revised by Gino Giotto, 5-Sep-2023.) Avoid ax-12 2171. (Revised by Rohan Ridenour, 9-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑥 𝑥 = 𝑦
 
8-Oct-2024rr-grothshort 41922 A shorter equivalent of ax-groth 10579 than rr-groth 41917 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 41921 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 44574 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 44573 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 44573 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 44567 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 44566 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 44564 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 44563 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 40117 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 40116 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 34964 Variant of sbievw 2095. (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 34963 Variant of equsexvw 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-equsalvwd 34962 Variant of equsalvw 2007. (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 34961 A variant of equsv 2006. (Contributed by BJ, 7-Oct-2024.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
 
7-Oct-2024bj-pm11.53a 34960 A variant of pm11.53v 1947. 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 34959 Version of pm11.53v 1947 with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024.)
((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
 
7-Oct-2024bj-pm11.53vw 34958 Version of pm11.53v 1947 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 40115 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 40114 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 40112 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 40111 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 40110 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 1857 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 34750 The negation of a theorem is equivalent to false. This can shorten dfnul2 4259. (Contributed by BJ, 5-Oct-2024.)
𝜑    ⇒   (¬ 𝜑 ↔ ⊥)
 
5-Oct-2024bj-mt2bi 34749 Version of mt2 199 where the major premise is a biconditional. Another proof is also possible via con2bii 358 and mpbi 229. The current mt2bi 364 should be relabeled, maybe to imfal. (Contributed by BJ, 5-Oct-2024.)
𝜑    &   (𝜓 ↔ ¬ 𝜑)    ⇒    ¬ 𝜓
 
5-Oct-2024sbc6g 3746 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 3609 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 2137, ax-11 2154, ax-12 2171. (Revised by SN, 5-Oct-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
 
5-Oct-2024elabg 3607 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 2137, ax-11 2154, ax-12 2171. (Revised by SN, 5-Oct-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
 
5-Oct-2024elab6g 3600 Membership in a class abstraction. Class version of sb6 2088. (Contributed by SN, 5-Oct-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
 
4-Oct-2024bj-gabima 35128 Generalized class abstraction as a direct image.

TODO: improve the support lemmas elimag 5973 and fvelima 6835 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 35127 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 35126 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 35125 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 35124 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 35123 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 35122 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 24863 Lemma for itg1add 24866. (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 7740 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 7749. (Contributed by NM, 18-Feb-2004.) Avoid ax-10 2137, ax-12 2171. (Revised by Gino Giotto, 3-Oct-2024.)
((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
 
3-Oct-2024dffr2 5553 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 2137, ax-11 2154, ax-12 2171, but use ax-8 2108. (Revised by Gino Giotto, 3-Oct-2024.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
 
3-Oct-2024pocl 5510 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 5480 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 4307 The class of sets verifying a property is the empty class if and only if that property is a contradiction. Version of ab0 4308 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 3517 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 3511 Implicit substitution of a class for a setvar variable. Version of vtocl3gf 3509 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 3130 Restricted specialization. Weak version of rsp 3131, requiring ax-8 2108, but not ax-12 2171. (Contributed by Gino Giotto, 3-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
 
1-Oct-2024thincfth 46329 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 46328 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 46327 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 46326 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 46293). (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 46325 Lemma for functhinc 46326. 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 46324 Lemma for functhinc 46326. 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 46323 Lemma for functhinc 46326. (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 46322 Lemma for functhinc 46326. 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 46299 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 46182 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 46181 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 46180 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 46179 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 46178 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 46177 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 46175 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 40109 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 40108 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 40107 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 46321 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 46290 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 46289 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 46288 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 46287 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 46286 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 46285 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 46284 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 46283 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 46280 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 46279 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 46278 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 7242 Change bound variable in a restricted description binder. Version of cbvriotav 7247 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 6399 Change bound variables in a description binder. Version of cbviotav 6402 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 4632 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 2137, ax-12 2171. (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 4630 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 2137, ax-12 2171. (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 4610 Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) Avoid ax-10 2137, ax-12 2171. (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
 
30-Sep-2024ralsng 4609 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 2137, ax-12 2171. (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
 
30-Sep-2024ralidmw 4438 Idempotent law for restricted quantifier. Weak version of ralidm 4442, which does not require ax-10 2137, ax-12 2171, but requires ax-8 2108. (Contributed by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
 
30-Sep-2024rabeq0w 4317 Condition for a restricted class abstraction to be empty. Version of rabeq0 4318 using implicit substitution, which does not require ax-10 2137, ax-11 2154, ax-12 2171, but requires ax-8 2108. (Contributed by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
 
30-Sep-2024cbvreuvw 3386 Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3390 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 3385 Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmov 3391 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 46320 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 46282 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 46281 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 46277 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 18133 is in quantified form. mrelatglb 18278 could potentially be shortened using this. See mrelatglbALT 46282. (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 46276 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 46275 Lemma for ipoglbdm 46276 and ipoglb 46277. (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 46271 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 46269 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 46253 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 46193 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 46163 Superclass of the greatest lower bound. A dual statement of ssintub 4897. (Contributed by Zhi Wang, 29-Sep-2024.)
∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴
 
29-Sep-2024focofob 44572 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 44571 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 44571 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 44570 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 44542 Conditions for a restriction to be an onto function. Part of fresf1o 30966. (Contributed by AV, 29-Sep-2024.)
((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–onto→𝐶)
 
29-Sep-2024f1imaenfi 8981 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 8800). (Contributed by BTernaryTau, 29-Sep-2024.)
((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ Fin) → (𝐹 “ 𝐶) ≈ 𝐶)
 
29-Sep-2024rescnvimafod 6951 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 6702 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 6701 Composition of onto functions. Generalisation of foco 6702. (Contributed by AV, 29-Sep-2024.)
((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵)
 
29-Sep-2024focnvimacdmdm 6700 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 46274 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 18132 is in quantified form. mrelatlub 18280 could potentially be shortened using this. See mrelatlubALT 46281. (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 46273 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 46272 Lemma for ipolubdm 46273 and ipolub 46274. (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 46270 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 46252 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 46148 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 40106 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 40105 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 40104 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 46267 Poset meet is idempotent. latmidm 18192 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 46266 Poset join is idempotent. latjidm 18180 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 46164 Two ways of expressing a collection of subsets as seen in df-ntr 22171, unimax 4877, and others (Contributed by Zhi Wang, 27-Sep-2024.)
(𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})
 
27-Sep-2024mpbiran4d 46143 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 40103 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 40102 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 34093 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 33691 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 33690 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 15299 Image of a limit under a continuous map, two-arg version. Originally a subproof of rlimcn2 15300. (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 46361 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 46268 A toset is a lattice. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝐾 ∈ Toset → 𝐾 ∈ Lat)
 
26-Sep-2024glbpr 46261 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 46260 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 46259 Lemma for glbprdm 46260 and glbpr 46261. (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 46258 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 46257 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 46256 Lemma for lubprdm 46257 and lubpr 46258. (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 46255 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 46254 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 46251 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 46250 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 40169 Alternative to wcdeq 3698 and df-cdeq 3699. This flattens the syntax representation ( wi ( weq vx vy ) wph ) to ( sn-wcdeq vx vy wph ), illustrating the comment of df-cdeq 3699. (Contributed by SN, 26-Sep-2024.) (New usage is discouraged.)
wff (𝑥 = 𝑦 → 𝜑)
 
26-Sep-2024fvpr2 7067 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 7065 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 7063 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 2053 If a formula is true, then it is true for at least one instance. This is to 19.8a 2174 what spw 2037 is to sp 2176. (Contributed by SN, 26-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (𝜑 → ∃𝑥𝜑)
 
25-Sep-2024postc 46363 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 46362 Alternate proof for postcpos 46361. (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 46342 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 46341 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 46296 A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 46365 for converting a monoid to a category. Lemma for bj-endmnd 35489. (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 46265 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 46264 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 46263 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 46262 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 34092 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 34091 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 34090 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 33988 Two elements of separated sets obey less than. Deduction form of ssltsepc 33987. (Contributed by Scott Fenton, 25-Sep-2024.)
(𝜑 → 𝐴 <<s 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → 𝑋 <s 𝑌)
 
25-Sep-2024f1domfi 8967 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 8760). (Contributed by BTernaryTau, 25-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
 
25-Sep-2024en2sn 8831 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) Avoid ax-pow 5288. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7588. (Revised by BTernaryTau, 25-Sep-2024.)
((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵})
 
25-Sep-2024f1dom2g 8757 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 8760 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 3339 Version of rexeqbidv 3337 with additional disjoint variable conditions, not requiring ax-8 2108 nor df-clel 2816. (Contributed by Wolf Lammen, 25-Sep-2024.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
 
25-Sep-2024nfnaew 2145 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 46370 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 46340 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 46339 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 46338 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 46316 In a thin category, all morphisms are epimorphisms. The converse does not hold. See grptcepi 46378. (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 46315 In a thin category, all morphisms are monomorphisms. The converse does not hold. See grptcmon 46377. (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 46314 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 46306 A thin category is a category (deduction form). (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    ⇒   (𝜑 → 𝐶 ∈ Cat)
 
24-Sep-2024mofsssn 46173 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 46142 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 34089 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 34088 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 33986 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 17810 (SetCat‘2o) is a category (provable from setccat 17800 and 2oex 8308) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 17809. Notably, the empty set ∅ is simultaneously an object (setc2obas 17809) , an identity morphism from ∅ to ∅ (setcid 17801 or thincid 46314) , and a non-identity morphism from ∅ to 1o. See cat1lem 17811 and cat1 17812 for a more general statement. This category is also thin (setc2othin 46337), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 46335 for more details on the "equivalence". (Contributed by Zhi Wang, 24-Sep-2024.)
𝐶 = (SetCat‘2o)    &   𝐻 = (Hom ‘𝐶)    ⇒   ∅ ∈ ((∅𝐻∅) ∩ (∅𝐻1o))
 
24-Sep-2024setc2obas 17809 ∅ and 1o are distinct objects in (SetCat‘2o). This combined with setc2ohom 17810 demonstrates that the category does not have pairwise disjoint hom-sets. See also df-cat 17377 and cat1 17812. (Contributed by Zhi Wang, 24-Sep-2024.)
𝐶 = (SetCat‘2o)    &   𝐵 = (Base‘𝐶)    ⇒   (∅ ∈ 𝐵 ∧ 1o ∈ 𝐵 ∧ 1o ≠ ∅)
 
24-Sep-2024en1uniel 8818 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7588. (Revised by BTernaryTau, 24-Sep-2024.)
(𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)
 
24-Sep-2024en1b 8813 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7588. (Revised by BTernaryTau, 24-Sep-2024.)
(𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})
 
24-Sep-2024eqsnuniex 5283 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 3155 Obsolete version of nfra2w 3154 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 3148 Deduction version of nfralw 3151. Version of nfrald 3150 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 46378 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 46377 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 46295 An alternate construction of the preorder induced by a category. See catprs2 46293 for details. See also catprsc 46294 for a different construction. The two constructions are different because df-cat 17377 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 46162 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 46160 "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 46152 Alternate proof for reutru 46151. (Contributed by Zhi Wang, 23-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
 
23-Sep-2024reutru 46151 Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
(∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
 
23-Sep-2024rmotru 46150 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 46149 Two ways of expressing "at least one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
(∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤)
 
23-Sep-2024dtrucor3 46144 An example of how ax-5 1913 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5359 in the ZF set theory. axc16nf 2255 and euae 2661 demonstrate that the violation of dtru 5359 leads to a model with only one object assuming its existence (ax-6 1971). The conclusion is also provable in the empty model ( see emptyal 1911). See also nf5 2279 and nf5i 2142 for the relation between unconditional ax-5 1913 and being not free. (Contributed by Zhi Wang, 23-Sep-2024.)
¬ ∀𝑥 𝑥 = 𝑦    &   (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)    ⇒   ∀𝑥 𝑥 = 𝑦
 
23-Sep-2024phplem1 8990 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. (Contributed by NM, 26-May-1998.) Avoid ax-pow 5288. (Revised by BTernaryTau, 23-Sep-2024.)
((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
 
23-Sep-2024entrfir 8977 Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8792). (Contributed by BTernaryTau, 23-Sep-2024.)
((𝐶 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
 
23-Sep-2024entrfi 8976 Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8792). (Contributed by BTernaryTau, 23-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
 
23-Sep-2024enfi 8973 Equinumerous sets have the same finiteness. For a shorter proof using ax-pow 5288, see enfiALT 8974. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5288. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
 
23-Sep-2024enfii 8972 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5288. (Revised by BTernaryTau, 23-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
 
23-Sep-2024ssnnfi 8952 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 8950 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.) Avoid ax-pow 5288. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ∈ ω → 𝐴 ∈ Fin)
 
23-Sep-2024en1 8811 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7588. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
 
23-Sep-2024ensn1 8807 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7588. (Revised by BTernaryTau, 23-Sep-2024.)
𝐴 ∈ V    ⇒   {𝐴} ≈ 1o
 
23-Sep-2024en0 8803 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5288, ax-un 7588. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ≈ ∅ ↔ 𝐴 = ∅)
 
23-Sep-2024bren 8743 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 8742 Equinumerosity relation. This variation of bren 8743 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 4264 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 2137, ax-11 2154, and ax-12 2171. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
¬ 𝐴 ∈ ∅
 
23-Sep-2024dfnul3 4260 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 4259 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2137, ax-11 2154, and ax-12 2171. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
 
23-Sep-2024dfnul4 4258 Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2108, df-clel 2816. (Revised by Gino Giotto, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4259. (Revised by BJ, 23-Sep-2024.)
∅ = {𝑥 ∣ ⊥}
 
23-Sep-2024nfabdw 2930 Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2932 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 46376 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 46375 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 46374 Lemma for mndtccat 46375 and mndtcid 46376. (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 46373 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 46372 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 46371 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 46369 Lemma for mndtchom 46371 and mndtcco 46372. (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 46368 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 46367 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 46366 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 46365 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 46367) , 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 46368, mndtchom 46371, mndtcco 46372. 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 35220 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 34918 Nonfreeness implies the equivalent of ax5ea 1916, inference form. (Contributed by BJ, 22-Sep-2024.)
Ⅎ'𝑥𝜑    ⇒   (∃𝑥𝜑 → ∀𝑥𝜑)
 
22-Sep-2024bj-nnfei 34915 Nonfreeness implies the equivalent of ax5e 1915, inference form. (Contributed by BJ, 22-Sep-2024.)
Ⅎ'𝑥𝜑    ⇒   (∃𝑥𝜑 → 𝜑)
 
22-Sep-2024bj-nnfai 34912 Nonfreeness implies the equivalent of ax-5 1913, inference form. See nf5ri 2188. (Contributed by BJ, 22-Sep-2024.)
Ⅎ'𝑥𝜑    ⇒   (𝜑 → ∀𝑥𝜑)
 
22-Sep-2024cphpyth 24380 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 3338 Version of raleqbidv 3336 with additional disjoint variable conditions, not requiring ax-8 2108 nor df-clel 2816. (Contributed by BJ, 22-Sep-2024.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
 
22-Sep-2024drnfc2 2928 Formula-building lemma for use with the Distinctor Reduction Theorem. Proof revision is marked as discouraged because the minimizer replaces albidv 1923 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 2108. (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 2926 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 2108, ax-11 2154. (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 1948 One direction of 19.36v 1991 that can be proven without ax-6 1971. (Contributed by Rohan Ridenour, 16-Apr-2022.) (Proof shortened by Wolf Lammen, 22-Sep-2024.)
(∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
 
21-Sep-2024prstchom2 46359 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 46360). 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 46313 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 46312 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 46311 Alternate proof for thincmo 46310. (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 46310 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 46298 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 46297 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 33067 Shorter proof of fineqvac 33066 using ax-rep 5209 and ax-pow 5288. (Contributed by BTernaryTau, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(Fin = V → CHOICE)
 
21-Sep-2024fineqvac 33066 If the Axiom of Infinity is negated, then the Axiom of Choice becomes redundant. For a shorter proof using ax-rep 5209 and ax-pow 5288, see fineqvacALT 33067. (Contributed by BTernaryTau, 21-Sep-2024.)
(Fin = V → CHOICE)
 
21-Sep-2024ffrnb 6615 Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6614. (Contributed by BJ, 21-Sep-2024.)
(𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
 
21-Sep-2024sbalex 2235 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 2068.

That both sides of the biconditional express proper substitution is proved by sb5 2268 and sb6 2088. The implication "to the left" is equs4v 2003 and does not require ax-10 2137 nor ax-12 2171. 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 2260 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 2068. (Revised by BJ, 21-Sep-2024.)

(∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
 
21-Sep-2024cad0 1620 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 46360 Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 46344. See prstchom2 46359 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 46358 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 46357 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 46356 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 46355 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 46354 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-2024prstcocvalOLD 46353 Obsolete proof of prstcocval 46352 as of 12-Nov-2024. Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ⊥ = (oc‘𝐾))    ⇒   (𝜑 → ⊥ = (oc‘𝐶))
 
20-Sep-2024prstcle 46351 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-2024prstclevalOLD 46350 Obsolete proof of prstcleval 46349 as of 12-Nov-2024. Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐾))    ⇒   (𝜑 → ≤ = (le‘𝐶))
 
20-Sep-2024prstcbas 46348 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 46347 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 46346 Lemma for prstcnid 46347 and prstchomval 46355. (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 46345 Lemma for prstcnidlem 46346 and prstcthin 46357. (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 46344 Definition of the function converting a preordered set to a category. Justified by prsthinc 46335.

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 46347, prstchom 46358, and prstcthin 46357. Other important properties include prstcbas 46348, prstcleval 46349, prstcle 46351, prstcocval 46352, prstcoc 46354, prstchom2 46359, and prstcprs 46356. Use those instead.

Note that the defining property prstchom 46358 is equivalent to prstchom2 46359 given prstcthin 46357. See thincn0eu 46313 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 46337 The category (SetCat‘2o) is thin. A special case of setcthin 46336. (Contributed by Zhi Wang, 20-Sep-2024.)
(SetCat‘2o) ∈ ThinCat
 
20-Sep-2024setcthin 46336 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 46184 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 46166 A variant of mof0 46165. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵)
 
20-Sep-2024f1co 6682 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 6633 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 6624 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 6622 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 6616 A function maps to its range iff the the range is a subset of its codomain. Generalization of ffrn 6614. (Contributed by AV, 20-Sep-2024.)
(𝜑 → ran 𝐹 ⊆ 𝐵)    ⇒   (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹))
 
20-Sep-2024fnco 6549 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 4137 Two ways of expressing that two classes have a given intersection. Inference form of ineqcom 4136. 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 1033 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 46333 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 46335, 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 46188 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 46187 assuming ax-un 7588 (see f1omoALT 46189). (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
 
19-Sep-2024mofmo 46174 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 46172 There is at most one function into a singleton. An unconditional variant of mofsn 46171, 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 46171 There is at most one function into a singleton, with fewer axioms than eufsn 46169 and eufsn2 46170. See also mofsn2 46172. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐵 ∈ 𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})
 
19-Sep-2024eufsn2 46170 There is exactly one function into a singleton, assuming ax-pow 5288 and ax-un 7588. Variant of eufsn 46169. If existence is not needed, use mofsn 46171 or mofsn2 46172 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 46169 There is exactly one function into a singleton, assuming ax-rep 5209. See eufsn2 46170 for different axiom requirements. If existence is not needed, use mofsn 46171 or mofsn2 46172 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 46168 There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 46169 assuming ax-rep 5209, or eufsn2 46170 assuming ax-pow 5288 and ax-un 7588. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉)    ⇒   (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
 
19-Sep-2024mof0ALT 46167 Alternate proof for mof0 46165 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 46165 There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
∃*𝑓 𝑓:𝐴⟶∅
 
19-Sep-2024mo0sn 46161 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 46159 "At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)
 
19-Sep-2024mosn 46158 "At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
 
19-Sep-2024vsn 46157 The singleton of the universal class is the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
{V} = ∅
 
19-Sep-2024f1cof1b 44569 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 8308 2o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2137, ax-11 2154, ax-12 2171, ax-un 7588. (Proof shortened by Zhi Wang, 19-Sep-2024.)
2o ∈ V
 
19-Sep-20241oex 8307 Ordinal 1 is a set. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by AV, 1-Jul-2022.) Remove dependency on ax-10 2137, ax-11 2154, ax-12 2171, ax-un 7588. (Revised by Zhi Wang, 19-Sep-2024.)
1o ∈ V
 
19-Sep-2024ecase2d 1027 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 46335 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 46292 and catprs2 46293 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 46294 A construction of the preorder induced by a category. See catprs2 46293 for details. See also catprsc2 46295 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 46293 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 46294 and catprsc2 46295 for constructions satisfying the hypothesis "catprs.1". See catprs 46292 for a more primitive version. See prsthinc 46335 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 46292 A preorder can be extracted from a category. See catprs2 46293 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 46291 Lemma for catprs 46292. (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 46249 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 46189 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 46188 without assuming ax-un 7588. (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 46187 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 46186 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 46185 Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   (𝜑 → 𝑋 ∈ (𝐴𝐹𝐵))    ⇒   (𝜑 → 𝐴𝑅𝐵)
 
18-Sep-2024fvconstr 46183 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 44568 Lemma for f1cof1b 44569 and focofob 44572. (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 44562 Lemma for fcoresf1 44563. (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 35042 Alternate proof of sbn1 2105, 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 34015 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 17396 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 6681 Composition of two one-to-one functions. Generalization of f1co 6682. (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 6623 Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 6624. (Contributed by AV, 18-Sep-2024.)
((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
 
18-Sep-2024cnvimassrndm 6055 The preimage of a superset of the range of a class is the domain of the class. Generalization of cnvimarndm 5990 for subsets. (Contributed by AV, 18-Sep-2024.)
(ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)
 
18-Sep-2024abeq2w 2815 Version of abeq2 2872 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 46334 An alternate proof for indthinc 46333 assuming more axioms including ax-pow 5288 and ax-un 7588. (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 46332 The empty category (see 0cat 17398) is thin. (Contributed by Zhi Wang, 17-Sep-2024.)
∅ ∈ ThinCat
 
17-Sep-20240thincg 46331 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 46319 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 46318 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 46317 Lemma for isthincd2 46319. (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 46309 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 46308 Lemma for isthincd2 46319 and thincmo2 46309. (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 46307 A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
ThinCat ⊆ Cat
 
17-Sep-2024thincc 46305 A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
(𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
 
17-Sep-2024isthinc3 46304 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 46303 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 46302 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 46301 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 44565 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 44561 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 44560 Lemma 4 for fcores 44561. (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 44558 Lemma 2 for fcores 44561. (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 44557 Lemma 1 for fcores 44561. (Contributed by AV, 17-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    ⇒   (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))
 
17-Sep-2024sltlpss 34087 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 34086 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 6944 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 6548 Composition of a function with domain and a function as a function with domain. Generalization of fnco 6549. (Contributed by AV, 17-Sep-2024.)
((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
 
16-Sep-2024neircl 46198 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 46176 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 40171 The right conjunct in the right hand side of the equivalence of isdomn 20565 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 40342 dvdssq 16272 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 40341 dvdsexpnn 40340 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 40327 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 40326 0 divides 0. (Contributed by SN, 15-Sep-2024.)
0 ∥ 0
 
15-Sep-2024syl3an12 40175 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 40172 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 17812 The definition of category df-cat 17377 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 17809 and setc2ohom 17810 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 17741 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 17811 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 17812. (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 16238 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 13808 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 8652 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 8651 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 8650 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 8643 for 𝐴 ∉ V, fset0 8642 for 𝐴 = ∅, and fsetex 8644 for 𝐵 ∈ V, see also fsetexb 8652. (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 8649 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 8648 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 44556 First version of proof for fsetprcnex 8650, 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 44555 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 44554 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 44553 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 44552 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 44559 Lemma 3 for fcores 44561. (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 44551 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 44550 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 44549 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 44548 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 44547 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 44546 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 44545 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 44544 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 44543 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 8642 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 8641 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 33065 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 33064 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 46247 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 33689 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 8971 Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8792). (Contributed by BTernaryTau, 10-Sep-2024.)
((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
 
9-Sep-2024seppcld 46223 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 46222 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 46221 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 46214 (𝐴(,]1) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
(0 ≤ 𝐴 → (𝐴(,]1) ∈ II)
 
9-Sep-2024i0oii 46213 (0[,)𝐴) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝐴 ≤ 1 → (0[,)𝐴) ∈ II)
 
9-Sep-2024iooii 46211 Open intervals are open sets of II. (Contributed by Zhi Wang, 9-Sep-2024.)
((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴(,)𝐵) ∈ II)
 
9-Sep-2024cnneiima 46210 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 46192 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 46191 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 46190 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 46156 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 33842 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 33841 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 33840 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 33839 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 33838 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 8970 Symmetry of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike ensymb 8788). (Contributed by BTernaryTau, 9-Sep-2024.)
(𝐴 ∈ Fin → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴))
 
9-Sep-2024f1oenfirn 8966 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 8963 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5288. (Revised by BTernaryTau, 9-Sep-2024.)
(𝐴 ∈ Fin → ◡𝐴 ∈ Fin)
 
9-Sep-2024f1dom3g 8755 The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 8760 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 46220 If two sets are separated by closed neighborhoods, then they are separated by (open) neighborhoods. See sepnsepo 46217 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 46198, adantr 481, and rexlimiva 3210. (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 46212 Closed intervals are closed sets of II. Note that iccss 13147, iccordt 22365, and ordtresticc 22374 are proved from ixxss12 13099, ordtcld3 22350, and ordtrest2 22355, respectively. An alternate proof uses restcldi 22324, dfii2 24045, and icccld 23930. (Contributed by Zhi Wang, 8-Sep-2024.)
((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴[,]𝐵) ∈ (Clsd‘II))
 
8-Sep-2024enreffi 8969 Equinumerosity is reflexive for finite sets, proved without using the Axiom of Power Sets (unlike enrefg 8772). (Contributed by BTernaryTau, 8-Sep-2024.)
(𝐴 ∈ Fin → 𝐴 ≈ 𝐴)
 
8-Sep-2024f1oenfi 8965 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 8759). (Contributed by BTernaryTau, 8-Sep-2024.)
((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
 
8-Sep-2024relopabv 5731 A class of ordered pairs is a relation. For a version without a disjoint variable condition, but using ax-11 2154 and ax-12 2171, see relopab 5734. (Contributed by SN, 8-Sep-2024.)
Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
8-Sep-2024ab0 4308 The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 4314 (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 2944). (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2816, ax-8 2108. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by SN, 8-Sep-2024.)
({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
8-Sep-2024ceqsralv 3469 Restricted quantifier version of ceqsalv 3467. (Contributed by NM, 21-Jun-2013.) Avoid ax-9 2116, ax-12 2171, ax-ext 2709. (Revised by SN, 8-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
 
8-Sep-2024ceqsalv 3467 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 2171. (Revised by SN, 8-Sep-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
 
7-Sep-2024seposep 46219 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 46217. The relationship between separatedness and closure is also seen in isnrm 22486, isnrm2 22509, isnrm3 22510. (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 46218 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 46154 Subset preserves disjointness. Deduction form of ssdisj 4393. Alternatively this could be proved with ineqcom 4136 in tandem with ssdisjd 46153. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → (𝐶 ∩ 𝐵) = ∅)    ⇒   (𝜑 → (𝐶 ∩ 𝐴) = ∅)
 
7-Sep-2024ssdisjd 46153 Subset preserves disjointness. Deduction form of ssdisj 4393. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → (𝐵 ∩ 𝐶) = ∅)    ⇒   (𝜑 → (𝐴 ∩ 𝐶) = ∅)
 
7-Sep-2024naddssim 33837 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 8961 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 5288. (Revised by BTernaryTau, 7-Sep-2024.)
(𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
 
7-Sep-2024pwfilem 8960 Lemma for pwfi 8961. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5288. (Revised by BTernaryTau, 7-Sep-2024.)
𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))    ⇒   (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
 
7-Sep-2024pwfir 8959 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 8958 Images of finite sets are finite. For a shorter proof using ax-pow 5288, see imafiALT 9112. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5288. (Revised by BTernaryTau, 7-Sep-2024.)
((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin)
 
6-Sep-2024clddisj 46197 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 46196 with elssuni 4871 replaced by the combination of cldss 22180 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 46196 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 46194 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 46147 A complex deduction form of r19.41v 3276. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))
 
6-Sep-2024ralbidb 46145 Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc 46146 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 46140 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 4338 Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.) Avoid ax-8 2108, df-clel 2816. (Revised by Gino Giotto, 6-Sep-2024.)
(𝜑 → ¬ 𝑥 ∈ 𝐴)    ⇒   (𝜑 → 𝐴 = ∅)
 
6-Sep-2024eq0 4277 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 2154, ax-12 2171. (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2108, df-clel 2816. (Revised by Gino Giotto, 6-Sep-2024.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
 
6-Sep-2024vn0 4272 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2108, df-clel 2816. (Revised by Gino Giotto, 6-Sep-2024.)
V ≠ ∅
 
5-Sep-2024iscnrm4 46248 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 46246 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 46245 Lemma for iscnrm3 46246. 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 46244 Lemma for iscnrm3l 46245. 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 42603.) (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 46242 Lemma for iscnrm3 46246. 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 46241 Lemma for iscnrm3r 46242. 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 46240 Lemma for iscnrm3rlem8 46241. 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 46239 Lemma for iscnrm3rlem7 46240. (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 46238 Lemma for iscnrm3rlem6 46239. (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 46237 Lemma for iscnrm3rlem8 46241. 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 46236 Lemma for iscnrm3r 46242. 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 46235 Lemma for iscnrm3rlem3 46236. (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 46234 Lemma for iscnrm3rlem2 46235. 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 46233 Lemma for iscnrm3rlem8 46241 and iscnrm3llem2 46244 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 46232 Lemma for iscnrm3lem7 46233. (Contributed by Zhi Wang, 5-Sep-2024.)
((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) ∧ 𝜓) → 𝜒)    ⇒   (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑊 𝜓 → 𝜒))
 
5-Sep-2024disjdifb 46155 Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.)
((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅
 
4-Sep-2024iscnrm3llem1 46243 Lemma for iscnrm3l 46245. 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 46230 Lemma for iscnrm3lem5 46231 and iscnrm3r 46242. (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 46229 Lemma for iscnrm3lem4 46230. (Contributed by Zhi Wang, 4-Sep-2024.)
(((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓))
 
4-Sep-2024on3ind 33829 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 33800 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 3436 All setvar variables are sets (see isset 3445). 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 2171. (Revised by SN, 28-Aug-2023.) (Proof shortened by BJ, 4-Sep-2024.)
𝑥 ∈ V
 
3-Sep-2024iscnrm3lem5 46231 Lemma for iscnrm3l 46245. (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 46228 Lemma for iscnrm3 46246 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 46227 Lemma for iscnrm3 46246. 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 46141 The import-export theorem (impexp 451) 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 33827 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 33826 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 46226 A topological space is normal if any disjoint closed sets can be separated by neighborhoods. An alternate definition of df-nrm 22468. (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 46209 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 46208 Lemma for restclssep 46209. (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 46207 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 46206 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 40167 Membership in a restricted class abstraction. This is to elrab2 3627 what elab2gw 40166 is to elab2g 3611. (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 4444 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 2816, ax-8 2108. (Revised by Gino Giotto, 2-Sep-2024.)
¬ 𝜑    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)
 
2-Sep-2024ral0 4443 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) Avoid df-clel 2816, ax-8 2108. (Revised by Gino Giotto, 2-Sep-2024.)
∀𝑥 ∈ ∅ 𝜑
 
2-Sep-2024ralidm 4442 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 4441 Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2816, ax-8 2108. (Revised by Gino Giotto, 2-Sep-2024.)
(∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
 
2-Sep-2024rzal 4439 Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid df-clel 2816, ax-8 2108. (Revised by Gino Giotto, 2-Sep-2024.)
(𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
 
2-Sep-2024sbc5 3744 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 3494 Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-10 2137, ax-11 2154, ax-12 2171. (Revised by SN, 2-Sep-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    &   (𝜑 → 𝜓)    ⇒   (𝜑 → 𝜒)
 
2-Sep-2024clelab 2883 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 2154, see sbc5ALT 3745 for more details. (Revised by SN, 2-Sep-2024.)
(𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
 
1-Sep-2024isnrm4 46224 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 46217 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 46216 Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 46217. Proof could be shortened by 1 step using ssdisjdr 46154. (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 46215 Lemma for sepnsepo 46217. (Contributed by Zhi Wang, 1-Sep-2024.)
(∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥 ∈ 𝐽 (𝜑 ∧ ∃𝑦 ∈ 𝐽 (𝜓 ∧ 𝜒)))
 
1-Sep-2024ruvALT 40168 Alternate proof of ruv 9361 with one fewer syntax step thanks to using elirrv 9355 instead of elirr 9356. 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 28764. (Contributed by SN, 1-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
{𝑥 ∣ 𝑥 ∉ 𝑥} = V
 
1-Sep-2024bj-clel3gALT 35221 Alternate proof of clel3g 3591. (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 5442 The law of concretion in terms of substitutions. Version of opelopabsb 5443 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 5407 Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) Use a similar proof to copsex4g 5409 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 4913 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 4912. (Revised by BJ, 1-Sep-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    ⇒   ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)
 
1-Sep-2024intprg 4912 The intersection of a pair is the intersection of its members. Closed form of intpr 4913. 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 4857 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 4856 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 4857 to prove it from uniprg 4856. (Revised by BJ, 1-Sep-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
 
1-Sep-2024clel4g 3593 Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent and avoid ax-12 2171. (Revised by BJ, 1-Sep-2024.)
(𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)))
 
1-Sep-2024clel2g 3588 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 2171. (Revised by BJ, 1-Sep-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)))
 
31-Aug-2024opnneieqvv 46205 The equivalence between neighborhood and open neighborhood. A variant of opnneieqv 46204 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 46204 The equivalence between neighborhood and open neighborhood. See opnneieqvv 46205 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 46203 A variant of opnneilv 46202. (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 46202 The converse of opnneir 46200 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 46198), 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 46201 A variant of opnneir 46200 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 46200 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 46199 Lemma factoring out common proof steps of opnneil 46203 and opnneirv 46201. (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 44584 Alternate definition of ℩': this is to df-aiota 44577 what dfiota4 6425 is to df-iota 6391. operation using the if operator. It is simpler than df-aiota 44577 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 44583 This is to df-aiota 44577 what iotauni 6408 is to df-iota 6391 (it uses intersection like df-aiota 44577, similar to iotauni 6408 using union like df-iota 6391; we could also prove an analogous result using union here too, in the same way that we have iotaint 6409). (Contributed by BJ, 31-Aug-2024.)
(∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
 
31-Aug-2024acos1half 40170 The arccosine of 1 / 2 is π / 3. (Contributed by SN, 31-Aug-2024.)
(arccos‘(1 / 2)) = (π / 3)
 
31-Aug-2024f1ofvswap 7178 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 46225 A topological space is normal if any disjoint closed sets can be separated by open neighborhoods. An alternate definition of df-nrm 22468. (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 46195 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 46146 Formula-building rule for restricted universal quantifier and additional condition (deduction form). A variant of ralbidb 46145. (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 46139 Distribution of implication over biconditional (deduction form). Variant of pm5.32da 579. (Contributed by Zhi Wang, 30-Aug-2024.)
((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜓)))
 
30-Aug-2024logic2 46138 Variant of logic1 46136. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    &   (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)))    ⇒   (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
 
30-Aug-2024logic1a 46137 Variant of logic1 46136. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    &   ((𝜑 ∧ 𝜓) → (𝜃 ↔ 𝜏))    ⇒   (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
 
30-Aug-2024logic1 46136 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 46135 Deduction converting a biconditional to a biconditional with conjunction. Variant of pm4.71d 562. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))
 
30-Aug-2024abn0 4314 Nonempty class abstraction. See also ab0 4308. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) Avoid df-clel 2816, ax-8 2108. (Revised by Gino Giotto, 30-Aug-2024.)
({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
 
30-Aug-2024ab0orv 4312 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 4309 Obsolete version of ab0 4308 as of 8-Sep-2024. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2816, ax-8 2108. (Revised by Gino Giotto, 30-Aug-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
30-Aug-2024rru 3714 Relative version of Russell's paradox ru 3715 (which corresponds to the case 𝐴 = V).

Originally a subproof in pwnss 5272. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3050. (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 3443 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 35091) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2816, ax-8 2108. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by BJ, 30-Aug-2024.)
({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)
 
29-Aug-2024dftermo3 17721 An alternate definition of df-termo 17700 depending on df-inito 17699, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.)
TermO = (InitO ∘ (oppCat ↾ Cat))
 
29-Aug-2024dfinito3 17720 An alternate definition of df-inito 17699 depending on df-termo 17700, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.)
InitO = (TermO ∘ (oppCat ↾ Cat))
 
29-Aug-2024dftermo2 17719 A terminal object is an initial object in the opposite category. An alternate definition of df-termo 17700 depending on df-inito 17699. (Contributed by Zhi Wang, 29-Aug-2024.)
TermO = (𝑐 ∈ Cat ↦ (InitO‘(oppCat‘𝑐)))
 
29-Aug-2024dfinito2 17718 An initial object is a terminal object in the opposite category. An alternate definition of df-inito 17699 depending on df-termo 17700. (Contributed by Zhi Wang, 29-Aug-2024.)
InitO = (𝑐 ∈ Cat ↦ (TermO‘(oppCat‘𝑐)))
 
29-Aug-2024zeroofn 17704 ZeroO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
ZeroO Fn Cat
 
29-Aug-2024termofn 17703 TermO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
TermO Fn Cat
 
29-Aug-2024initofn 17702 InitO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
InitO Fn Cat
 
29-Aug-2024oppccatf 17439 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 23746 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 23747 proved from this theorem and grpidcl 18607) or more generally monoids (see mndidcl 18400), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 23747. (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 33836 Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
(𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
 
26-Aug-2024naddcom 33835 Natural addition commutes. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))

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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