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	nrelv 5801	The universal class is not a relation. (Contributed by Thierry Arnoux, 23-Jan-2022.)
⊢ ¬ Rel V
Theorem	relsng 5802	A singleton is a relation iff it is a singleton on an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.)
⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
Theorem	relsnb 5803	An at-most-singleton is a relation iff it is empty (because it is a "singleton on a proper class") or it is a singleton of an ordered pair. (Contributed by BJ, 26-Feb-2023.)
⊢ (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
Theorem	relsnopg 5804	A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by BJ, 12-Feb-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {⟨𝐴, 𝐵⟩})
Theorem	relsn 5805	A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
Theorem	relsnop 5806	A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ Rel {⟨𝐴, 𝐵⟩}
Theorem	copsex2gb 5807*	Implicit substitution inference for ordered pairs. Compare copsex2ga 5808. (Contributed by NM, 12-Mar-2014.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
Theorem	copsex2ga 5808*	Implicit substitution inference for ordered pairs. Compare copsex2g 5494. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
Theorem	elopaba 5809*	Membership in an ordered-pair class abstraction. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
Theorem	xpsspw 5810	A Cartesian product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)
⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)
Theorem	unixpss 5811	The double class union of a Cartesian product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)
⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ (𝐴 ∪ 𝐵)
Theorem	relun 5812	The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)
⊢ (Rel (𝐴 ∪ 𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))
Theorem	relin1 5813	The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
⊢ (Rel 𝐴 → Rel (𝐴 ∩ 𝐵))
Theorem	relin2 5814	The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
⊢ (Rel 𝐵 → Rel (𝐴 ∩ 𝐵))
Theorem	relinxp 5815	Intersection with a Cartesian product is a relation. (Contributed by Peter Mazsa, 4-Mar-2019.)
⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
Theorem	reldif 5816	A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
⊢ (Rel 𝐴 → Rel (𝐴 ∖ 𝐵))
Theorem	reliun 5817	An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)
⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵)
Theorem	reliin 5818	An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)
⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵)
Theorem	reluni 5819*	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 5820*	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 5821*	A class of ordered pairs is a relation. For a version without a disjoint variable condition, but a longer proof using ax-11 2155 and ax-12 2172, see relopabi 5823. (Contributed by BJ, 22-Jul-2023.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopabv 5822*	A class of ordered pairs is a relation. For a version without a disjoint variable condition, but using ax-11 2155 and ax-12 2172, see relopab 5825. (Contributed by SN, 8-Sep-2024.)
⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	relopabi 5823	A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 5300, ax-nul 5307, ax-pr 5428. (Revised by KP, 25-Oct-2021.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopabiALT 5824	Alternate proof of relopabi 5823 (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 5825	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 5826	The maps-to notation always describes a binary relation. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	reli 5827	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 5828	The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
⊢ Rel E
Theorem	opabid2 5829*	A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
⊢ (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)
Theorem	inopab 5830*	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 5831*	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	difopabOLD 5832*	Obsolete version of difopab 5831 as of 19-Dec-2024. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
Theorem	inxp 5833	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.)
⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))
Theorem	xpindi 5834	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 5835	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 5836*	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 5837*	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 5838*	Membership in a union of Cartesian products. Analogue of elxp 5700 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 5839*	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 5840*	Write a double restricted quantification as one universal quantifier. In this version of ralxp 5842, 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 5841*	Write a double restricted quantification as one universal quantifier. In this version of rexxp 5843, 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 5842*	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 5843*	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	exopxfr 5844*	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 5845*	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 5846*	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 5847*	Version of ralxp 5842 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 5848*	Version of rexxp 5843 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 5849*	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 5850*	Deduce equality of a relation and an ordered-pair class abstraction. Compare eqabdv 2868. (Contributed by NM, 24-Feb-2014.)
⊢ Rel 𝐴    &   ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝜓))    ⇒   ⊢ (𝜑 → 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Theorem	relop 5851*	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 5804, as funsng 6600 is to funop 7147. (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
Theorem	ideqg 5852	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 5853	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 5854	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 5855	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 5856	Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶))
Theorem	coss2 5857	Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵))
Theorem	coeq1 5858	Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
Theorem	coeq2 5859	Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
Theorem	coeq1i 5860	Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)
Theorem	coeq2i 5861	Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)
Theorem	coeq1d 5862	Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
Theorem	coeq2d 5863	Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
Theorem	coeq12i 5864	Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)
Theorem	coeq12d 5865	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 5866	Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵)
Theorem	brcog 5867*	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 5868*	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 5869	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	eqbrrdva 5870*	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 5871*	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 5872*	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 5873	Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Kyle Wyonch, 27-Apr-2021.)
⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)
Theorem	cnveq 5874	Equality theorem for converse relation. (Contributed by NM, 13-Aug-1995.)
⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵)
Theorem	cnveqi 5875	Equality inference for converse relation. (Contributed by NM, 23-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ◡𝐴 = ◡𝐵
Theorem	cnveqd 5876	Equality deduction for converse relation. (Contributed by NM, 6-Dec-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ◡𝐴 = ◡𝐵)
Theorem	elcnv 5877*	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 5878*	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 5879	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 5880	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 5881	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 5882	Ordered-pair membership in converse relation. (Contributed by NM, 13-Aug-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ ◡𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
Theorem	brcnv 5883	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	csbcnv 5884	Move class substitution in and out of the converse of a relation. Version of csbcnvgALT 5885 without a sethood antecedent but depending on more axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (Revised by NM, 23-Aug-2018.)
⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹
Theorem	csbcnvgALT 5885	Move class substitution in and out of the converse of a relation. Version of csbcnv 5884 with a sethood antecedent but depending on fewer axioms. (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 5886	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 5887*	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 5888*	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 5889*	Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Theorem	dfrn3 5890*	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 5891*	Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵))
Theorem	elrng 5892*	Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))
Theorem	elrn2 5893*	Membership in a range. (Contributed by NM, 10-Jul-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵)
Theorem	elrn 5894*	Membership in a range. (Contributed by NM, 2-Apr-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)
Theorem	ssrelrn 5895*	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 5896	Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
⊢ dom 𝐴 = ran ◡𝐴
Theorem	dfdmf 5897*	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 5898	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 5899*	Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
Theorem	eldm2g 5900*	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))