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	releqi 5801	Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (Rel 𝐴 ↔ Rel 𝐵)
Theorem	releqd 5802	Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Rel 𝐴 ↔ Rel 𝐵))
Theorem	nfrel 5803	Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥Rel 𝐴
Theorem	sbcrel 5804	Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel ⦋𝐴 / 𝑥⦌𝑅))
Theorem	relss 5805	Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)
⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴))
Theorem	ssrel 5806*	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 5317, ax-nul 5324, ax-pr 5447. (Revised by KP, 25-Oct-2021.) Remove dependency on ax-12 2178. (Revised by SN, 11-Dec-2024.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Theorem	ssrelOLD 5807*	Obsolete version of ssrel 5806 as of 11-Dec-2024. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Remove dependency on ax-sep 5317, ax-nul 5324, ax-pr 5447. (Revised by KP, 25-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Theorem	eqrel 5808*	Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Theorem	ssrel2 5809*	A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 5806 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)
⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅 ⊆ 𝑆 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
Theorem	ssrel3 5810*	Subclass relation in another form when the subclass is a relation. (Contributed by Peter Mazsa, 16-Feb-2019.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦)))
Theorem	relssi 5811*	Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
⊢ Rel 𝐴    &   ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	relssdv 5812*	Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)
⊢ (𝜑 → Rel 𝐴)    &   ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	eqrelriv 5813*	Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)
⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)    ⇒   ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
Theorem	eqrelriiv 5814*	Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)
⊢ Rel 𝐴    &   ⊢ Rel 𝐵    &   ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)    ⇒   ⊢ 𝐴 = 𝐵
Theorem	eqbrriv 5815*	Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
⊢ Rel 𝐴    &   ⊢ Rel 𝐵    &   ⊢ (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)    ⇒   ⊢ 𝐴 = 𝐵
Theorem	eqrelrdv 5816*	Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)
⊢ Rel 𝐴    &   ⊢ Rel 𝐵    &   ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	eqbrrdv 5817*	Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ (𝜑 → Rel 𝐴)    &   ⊢ (𝜑 → Rel 𝐵)    &   ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	eqbrrdiv 5818*	Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)
⊢ Rel 𝐴    &   ⊢ Rel 𝐵    &   ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	eqrelrdv2 5819*	A version of eqrelrdv 5816. (Contributed by Rodolfo Medina, 10-Oct-2010.)
⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))    ⇒   ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Theorem	ssrelrel 5820*	A subclass relationship determined by ordered triples. Use relrelss 6304 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ⊆ ((V × V) × V) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
Theorem	eqrelrel 5821*	Extensionality principle for ordered triples (used by 2-place operations df-oprab 7452), analogous to eqrel 5808. Use relrelss 6304 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.)
⊢ ((𝐴 ∪ 𝐵) ⊆ ((V × V) × V) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
Theorem	elrel 5822*	A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Theorem	rel0 5823	The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
⊢ Rel ∅
Theorem	nrelv 5824	The universal class is not a relation. (Contributed by Thierry Arnoux, 23-Jan-2022.)
⊢ ¬ Rel V
Theorem	relsng 5825	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 5826	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 5827	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 5828	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 5829	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 5830*	Implicit substitution inference for ordered pairs. Compare copsex2ga 5831. (Contributed by NM, 12-Mar-2014.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
Theorem	copsex2ga 5831*	Implicit substitution inference for ordered pairs. Compare copsex2g 5513. (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 5832*	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 5833	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 5834	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 5835	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 5836	The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
⊢ (Rel 𝐴 → Rel (𝐴 ∩ 𝐵))
Theorem	relin2 5837	The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
⊢ (Rel 𝐵 → Rel (𝐴 ∩ 𝐵))
Theorem	relinxp 5838	Intersection with a Cartesian product is a relation. (Contributed by Peter Mazsa, 4-Mar-2019.)
⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
Theorem	reldif 5839	A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
⊢ (Rel 𝐴 → Rel (𝐴 ∖ 𝐵))
Theorem	reliun 5840	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 5841	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 5842*	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 5843*	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 5844*	A class of ordered pairs is a relation. For a version without a disjoint variable condition, but a longer proof using ax-11 2158 and ax-12 2178, see relopabi 5846. (Contributed by BJ, 22-Jul-2023.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopabv 5845*	A class of ordered pairs is a relation. For a version without a disjoint variable condition, but using ax-11 2158 and ax-12 2178, see relopab 5848. (Contributed by SN, 8-Sep-2024.)
⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	relopabi 5846	A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 5317, ax-nul 5324, ax-pr 5447. (Revised by KP, 25-Oct-2021.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopabiALT 5847	Alternate proof of relopabi 5846 (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 5848	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 5849	The maps-to notation always describes a binary relation. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	reli 5850	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 5851	The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
⊢ Rel E
Theorem	opabid2 5852*	A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
⊢ (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)
Theorem	inopab 5853*	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 5854*	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 5855*	Obsolete version of difopab 5854 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 5856	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 2141, ax-12 2178. (Revised by SN, 5-May-2025.)
⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))
Theorem	inxpOLD 5857	Obsolete version of inxp 5856 as of 5-May-2025. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))
Theorem	xpindi 5858	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 5859	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 5860*	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 5861*	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 5862*	Membership in a union of Cartesian products. Analogue of elxp 5723 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 5863*	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 5864*	Write a double restricted quantification as one universal quantifier. In this version of ralxp 5866, 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 5865*	Write a double restricted quantification as one universal quantifier. In this version of rexxp 5867, 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 5866*	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 5867*	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 5868*	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 5869*	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 5870*	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 5871*	Version of ralxp 5866 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 5872*	Version of rexxp 5867 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 5873*	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 5874*	Deduce equality of a relation and an ordered-pair class abstraction. Compare eqabdv 2878. (Contributed by NM, 24-Feb-2014.)
⊢ Rel 𝐴    &   ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝜓))    ⇒   ⊢ (𝜑 → 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Theorem	relop 5875*	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 5827, as funsng 6629 is to funop 7183. (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
Theorem	ideqg 5876	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 5877	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 5878	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 5879	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 5880	Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶))
Theorem	coss2 5881	Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵))
Theorem	coeq1 5882	Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
Theorem	coeq2 5883	Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
Theorem	coeq1i 5884	Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)
Theorem	coeq2i 5885	Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)
Theorem	coeq1d 5886	Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
Theorem	coeq2d 5887	Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
Theorem	coeq12i 5888	Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)
Theorem	coeq12d 5889	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 5890	Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵)
Theorem	brcog 5891*	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 5892*	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 5893	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 5894*	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 5895*	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 5896*	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 5897	Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Kyle Wyonch, 27-Apr-2021.)
⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)
Theorem	cnveq 5898	Equality theorem for converse relation. (Contributed by NM, 13-Aug-1995.)
⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵)
Theorem	cnveqi 5899	Equality inference for converse relation. (Contributed by NM, 23-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ◡𝐴 = ◡𝐵
Theorem	cnveqd 5900	Equality deduction for converse relation. (Contributed by NM, 6-Dec-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ◡𝐴 = ◡𝐵)