P. 59 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 5801-5900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	reluni 5801*	The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by NM, 13-Aug-2004.)
⊢ (Rel ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥)
Theorem	relint 5802*	The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)
⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝐴)
Theorem	relopabiv 5803*	Alternate proof of relopabi 5805, which is shorter but requires an extra DV condition. (Contributed by BJ, 22-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopabvOLD 5804*	Obsolete version of relopab 5808 as of 26-Oct-2026. (Contributed by SN, 8-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	relopabi 5805	A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 5252, ax-nul 5263, ax-pr 5395. (Revised by KP, 25-Oct-2021.) Avoid ax-11 2197 and ax-12 2216. (Revised by SN, 26-Oct-2026.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopabiOLD 5806	Obsolete version of relopabi 5805 as of 26-Oct-2026. (Contributed by Mario Carneiro, 21-Dec-2013.) (Revised by KP, 25-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopabiALT 5807	Alternate proof of relopabi 5805 (shorter but uses more axioms). (Contributed by Mario Carneiro, 21-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopab 5808	A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) Remove disjoint variable conditions. (Revised by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)
⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	mptrel 5809	The maps-to notation always describes a binary relation. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	reli 5810	The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
⊢ Rel I
Theorem	rele 5811	The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
⊢ Rel E
Theorem	opabid2 5812*	A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
⊢ (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)
Theorem	inopab 5813*	Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
Theorem	difopab 5814*	Difference of two ordered-pair class abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Proof shortened by SN, 19-Dec-2024.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
Theorem	inxp 5815	Intersection of two Cartesian products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2180, ax-12 2216. (Revised by SN, 5-May-2025.)
⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))
Theorem	xpindi 5816	Distributive law for Cartesian product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
⊢ (𝐴 × (𝐵 ∩ 𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))
Theorem	xpindir 5817	Distributive law for Cartesian product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
⊢ ((𝐴 ∩ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))
Theorem	xpiindi 5818*	Distributive law for Cartesian product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐴 ≠ ∅ → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
Theorem	xpriindi 5819*	Distributive law for Cartesian product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
Theorem	eliunxp 5820*	Membership in a union of Cartesian products. Analogue of elxp 5678 for nonconstant 𝐵(𝑥). (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
Theorem	opeliunxp2 5821*	Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 14-Feb-2015.)
⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸)    ⇒   ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))
Theorem	raliunxp 5822*	Write a double restricted quantification as one universal quantifier. In this version of ralxp 5824, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)
Theorem	rexiunxp 5823*	Write a double restricted quantification as one universal quantifier. In this version of rexxp 5825, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)
Theorem	ralxp 5824*	Universal quantification restricted to a Cartesian product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)
Theorem	rexxp 5825*	Existential quantification restricted to a Cartesian product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)
Theorem	el2xptp 5826*	A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
Theorem	exopxfr 5827*	Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦∃𝑧𝜓)
Theorem	exopxfr2 5828*	Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴 ∧ 𝜓)))
Theorem	djussxp 5829*	Disjoint union is a subset of a Cartesian product. (Contributed by Stefan O'Rear, 21-Nov-2014.)
⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)
Theorem	ralxpf 5830*	Version of ralxp 5824 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)
Theorem	rexxpf 5831*	Version of rexxp 5825 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)
Theorem	iunxpf 5832*	Indexed union on a Cartesian product equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)
⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑧𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)    ⇒   ⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷
Theorem	opabbi2dv 5833*	Deduce equality of a relation and an ordered-pair class abstraction. Compare eqabdv 2897. (Contributed by NM, 24-Feb-2014.)
⊢ Rel 𝐴    &   ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝜓))    ⇒   ⊢ (𝜑 → 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Theorem	relop 5834*	A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) A relation is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a relation is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is relsnopg 5786, as funsng 6613 is to funop 7177. (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
Theorem	ideqg 5835	For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	ideq 5836	For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)
Theorem	ididg 5837	A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴)
Theorem	issetid 5838	Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 ∈ V ↔ 𝐴 I 𝐴)
Theorem	coss1 5839	Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶))
Theorem	coss2 5840	Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵))
Theorem	coeq1 5841	Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
Theorem	coeq2 5842	Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
Theorem	coeq1i 5843	Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)
Theorem	coeq2i 5844	Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)
Theorem	coeq1d 5845	Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
Theorem	coeq2d 5846	Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
Theorem	coeq12i 5847	Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)
Theorem	coeq12d 5848	Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷))
Theorem	nfco 5849	Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵)
Theorem	brcog 5850*	Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))
Theorem	opelco2g 5851*	Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐷 ∧ ⟨𝑥, 𝐵⟩ ∈ 𝐶)))
Theorem	brcogw 5852	Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝑍) ∧ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)) → 𝐴(𝐶 ∘ 𝐷)𝐵)
Theorem	brcoi 5853	Ordered pair membership in a composition. (Contributed by BJ, 25-Oct-2026.)
⊢ 𝑋 ∈ V    &   ⊢ 𝑌 ∈ V    &   ⊢ 𝑍 ∈ V    ⇒   ⊢ ((𝑋𝑅𝑌 ∧ 𝑌𝑆𝑍) → 𝑋(𝑆 ∘ 𝑅)𝑍)
Theorem	eqbrrdva 5854*	Deduction from extensionality principle for relations, given an equivalence only on the relation domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
⊢ (𝜑 → 𝐴 ⊆ (𝐶 × 𝐷))    &   ⊢ (𝜑 → 𝐵 ⊆ (𝐶 × 𝐷))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	brco 5855*	Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))
Theorem	opelco 5856*	Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))
Theorem	cnvss 5857	Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Kyle Wyonch, 27-Apr-2021.)
⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)
Theorem	cnveq 5858	Equality theorem for converse relation. (Contributed by NM, 13-Aug-1995.)
⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵)
Theorem	cnveqi 5859	Equality inference for converse relation. (Contributed by NM, 23-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ◡𝐴 = ◡𝐵
Theorem	cnveqd 5860	Equality deduction for converse relation. (Contributed by NM, 6-Dec-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ◡𝐴 = ◡𝐵)
Theorem	elcnv 5861*	Membership in a converse relation. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
Theorem	elcnv2 5862*	Membership in a converse relation. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
Theorem	nfcnv 5863	Bound-variable hypothesis builder for converse relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥◡𝐴
Theorem	brcnvg 5864	The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))
Theorem	opelcnvg 5865	Ordered-pair membership in converse relation. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ ◡𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
Theorem	opelcnv 5866	Ordered-pair membership in converse relation. (Contributed by NM, 13-Aug-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ ◡𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
Theorem	brcnv 5867	The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)
Theorem	cnv0 5868	The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5252, ax-nul 5263, ax-pr 5395. (Revised by KP, 25-Oct-2021.) Avoid ax-12 2216. (Revised by TM, 31-Jan-2026.)
⊢ ◡∅ = ∅
Theorem	cnv0OLD 5869	Obsolete version of cnv0 5868 as of 31-Jan-2026. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5252, ax-nul 5263, ax-pr 5395. (Revised by KP, 25-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ◡∅ = ∅
Theorem	cnvi 5870	The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ◡ I = I
Theorem	csbcnv 5871	Move class substitution in and out of the converse of a relation. (Contributed by Thierry Arnoux, 8-Feb-2017.) (Revised by NM, 23-Aug-2018.) Remove dependency on ax-sep 5252 and ax-pr 5395. (Revised by Eric Schmidt, 4-Jun-2026.)
⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹
Theorem	csbcnvOLD 5872	Obsolete version of csbcnv 5871 as of 4-Jun-2026. (Contributed by Thierry Arnoux, 8-Feb-2017.) (Revised by NM, 23-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹
Theorem	csbcnvgALTOLD 5873	Obsolete version of csbcnv 5871 as of 4-Jun-2026. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹)
Theorem	cnvco 5874	Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
Theorem	cnvuni 5875*	The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)
⊢ ◡∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ◡𝑥
Theorem	dfdm3 5876*	Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴}
Theorem	dfrn2 5877*	Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Theorem	dfrn3 5878*	Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
Theorem	elrn2g 5879*	Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵))
Theorem	elrng 5880*	Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))
Theorem	elrn2 5881*	Membership in a range. (Contributed by NM, 10-Jul-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵)
Theorem	elrn 5882*	Membership in a range. (Contributed by NM, 2-Apr-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)
Theorem	ssrelrn 5883*	If a relation is a subset of a cartesian product, then for each element of the range of the relation there is an element of the first set of the cartesian product which is related to the element of the range by the relation. (Contributed by AV, 24-Oct-2020.)
⊢ ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎 ∈ 𝐴 𝑎𝑅𝑌)
Theorem	dfdm4 5884	Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
⊢ dom 𝐴 = ran ◡𝐴
Theorem	dfdmf 5885*	Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    ⇒   ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
Theorem	csbdm 5886	Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.) (Revised by NM, 24-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵
Theorem	eldmg 5887*	Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
Theorem	eldm2g 5888*	Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐵))
Theorem	eldm 5889*	Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)
Theorem	eldm2 5890*	Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐵)
Theorem	dmss 5891	Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
⊢ (𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵)
Theorem	dmeq 5892	Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
Theorem	dmeqi 5893	Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ dom 𝐴 = dom 𝐵
Theorem	dmeqd 5894	Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → dom 𝐴 = dom 𝐵)
Theorem	opeldmd 5895	Membership of first of an ordered pair in a domain. Deduction version of opeldm 5896. (Contributed by AV, 11-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶))
Theorem	opeldm 5896	Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶)
Theorem	breldm 5897	Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)
Theorem	breldmg 5898	Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
Theorem	dmun 5899	The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵)
Theorem	dmin 5900	The domain of an intersection is included in the intersection of the domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
⊢ dom (𝐴 ∩ 𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)