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Theorem List for Intuitionistic Logic Explorer - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssrel2 4701* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4699 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)
(𝑅 ⊆ (𝐴 × 𝐵) → (𝑅𝑆 ↔ ∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
 
Theoremrelssi 4702* Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
Rel 𝐴    &   (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)       𝐴𝐵
 
Theoremrelssdv 4703* Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)
(𝜑 → Rel 𝐴)    &   (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (𝜑𝐴𝐵)
 
Theoremeqrelriv 4704* Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)
(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)       ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
 
Theoremeqrelriiv 4705* Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)
Rel 𝐴    &   Rel 𝐵    &   (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)       𝐴 = 𝐵
 
Theoremeqbrriv 4706* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
Rel 𝐴    &   Rel 𝐵    &   (𝑥𝐴𝑦𝑥𝐵𝑦)       𝐴 = 𝐵
 
Theoremeqrelrdv 4707* 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))       (𝜑𝐴 = 𝐵)
 
Theoremeqbrrdv 4708* Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑 → Rel 𝐴)    &   (𝜑 → Rel 𝐵)    &   (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)
 
Theoremeqbrrdiv 4709* Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)
Rel 𝐴    &   Rel 𝐵    &   (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)
 
Theoremeqrelrdv2 4710* A version of eqrelrdv 4707. (Contributed by Rodolfo Medina, 10-Oct-2010.)
(𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
 
Theoremssrelrel 4711* A subclass relationship determined by ordered triples. Use relrelss 5137 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) → (𝐴𝐵 ↔ ∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
 
Theoremeqrelrel 4712* Extensionality principle for ordered triples, analogous to eqrel 4700. Use relrelss 5137 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.)
((𝐴𝐵) ⊆ ((V × V) × V) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
 
Theoremelrel 4713* A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
 
Theoremrelsng 4714 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.)
(𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
 
Theoremrelsnopg 4715 A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by BJ, 12-Feb-2022.)
((𝐴𝑉𝐵𝑊) → Rel {⟨𝐴, 𝐵⟩})
 
Theoremrelsn 4716 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)
𝐴 ∈ V       (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
 
Theoremrelsnop 4717 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⟩}
 
Theoremxpss12 4718 Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷))
 
Theoremxpss 4719 A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
(𝐴 × 𝐵) ⊆ (V × V)
 
Theoremrelxp 4720 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)
Rel (𝐴 × 𝐵)
 
Theoremxpss1 4721 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
(𝐴𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))
 
Theoremxpss2 4722 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
(𝐴𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵))
 
Theoremxpsspw 4723 A cross 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 (𝐴𝐵)
 
Theoremunixpss 4724 The double class union of a cross product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)
(𝐴 × 𝐵) ⊆ (𝐴𝐵)
 
Theoremxpexg 4725 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)
((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
 
Theoremxpex 4726 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 × 𝐵) ∈ V
 
Theoremsqxpexg 4727 The Cartesian square of a set is a set. (Contributed by AV, 13-Jan-2020.)
(𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
 
Theoremrelun 4728 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)
(Rel (𝐴𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))
 
Theoremrelin1 4729 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
(Rel 𝐴 → Rel (𝐴𝐵))
 
Theoremrelin2 4730 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
(Rel 𝐵 → Rel (𝐴𝐵))
 
Theoremreldif 4731 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
(Rel 𝐴 → Rel (𝐴𝐵))
 
Theoremreliun 4732 An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)
(Rel 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 Rel 𝐵)
 
Theoremreliin 4733 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.)
(∃𝑥𝐴 Rel 𝐵 → Rel 𝑥𝐴 𝐵)
 
Theoremreluni 4734* 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 ↔ ∀𝑥𝐴 Rel 𝑥)
 
Theoremrelint 4735* The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)
(∃𝑥𝐴 Rel 𝑥 → Rel 𝐴)
 
Theoremrel0 4736 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
Rel ∅
 
Theoremrelopabi 4737 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       Rel 𝐴
 
Theoremrelopab 4738 A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) (Unnecessary distinct variable restrictions were removed by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)
Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theoremmptrel 4739 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
Rel (𝑥𝐴𝐵)
 
Theoremreli 4740 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
 
Theoremrele 4741 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
Rel E
 
Theoremopabid2 4742* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
(Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)
 
Theoreminopab 4743* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}
 
Theoremdifopab 4744* The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
 
Theoreminxp 4745 The intersection of two cross 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)) = ((𝐴𝐶) × (𝐵𝐷))
 
Theoremxpindi 4746 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
(𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))
 
Theoremxpindir 4747 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))
 
Theoremxpiindim 4748* Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
(∃𝑦 𝑦𝐴 → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
 
Theoremxpriindim 4749* Distributive law for cross product over relativized indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
(∃𝑦 𝑦𝐴 → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
 
Theoremeliunxp 4750* Membership in a union of cross products. Analogue of elxp 4628 for nonconstant 𝐵(𝑥). (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
 
Theoremopeliunxp2 4751* Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)
(𝑥 = 𝐶𝐵 = 𝐸)       (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
 
Theoremraliunxp 4752* Write a double restricted quantification as one universal quantifier. In this version of ralxp 4754, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))       (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
 
Theoremrexiunxp 4753* Write a double restricted quantification as one universal quantifier. In this version of rexxp 4755, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)
(𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))       (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
 
Theoremralxp 4754* Universal quantification restricted to a cross 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⟩ → (𝜑𝜓))       (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
 
Theoremrexxp 4755* Existential quantification restricted to a cross product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)
(𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))       (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
 
Theoremdjussxp 4756* Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)
𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)
 
Theoremralxpf 4757* Version of ralxp 4754 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   𝑧𝜑    &   𝑥𝜓    &   (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))       (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
 
Theoremrexxpf 4758* Version of rexxp 4755 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   𝑧𝜑    &   𝑥𝜓    &   (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))       (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
 
Theoremiunxpf 4759* Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)
𝑦𝐶    &   𝑧𝐶    &   𝑥𝐷    &   (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)        𝑥 ∈ (𝐴 × 𝐵)𝐶 = 𝑦𝐴 𝑧𝐵 𝐷
 
Theoremopabbi2dv 4760* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2289. (Contributed by NM, 24-Feb-2014.)
Rel 𝐴    &   (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝜓))       (𝜑𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
 
Theoremrelop 4761* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
𝐴 ∈ V    &   𝐵 ∈ V       (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
 
Theoremideqg 4762 For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
 
Theoremideq 4763 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
𝐵 ∈ V       (𝐴 I 𝐵𝐴 = 𝐵)
 
Theoremididg 4764 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝑉𝐴 I 𝐴)
 
Theoremissetid 4765 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)
 
Theoremcoss1 4766 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremcoss2 4767 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremcoeq1 4768 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremcoeq2 4769 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremcoeq1i 4770 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremcoeq2i 4771 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremcoeq1d 4772 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremcoeq2d 4773 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremcoeq12i 4774 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)
 
Theoremcoeq12d 4775 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 
Theoremnfco 4776 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)
 
Theoremelco 4777* Elements of a composed relation. (Contributed by BJ, 10-Jul-2022.)
(𝐴 ∈ (𝑅𝑆) ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦𝑦𝑅𝑧)))
 
Theorembrcog 4778* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
 
Theoremopelco2g 4779* Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐷 ∧ ⟨𝑥, 𝐵⟩ ∈ 𝐶)))
 
Theorembrcogw 4780 Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
(((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)
 
Theoremeqbrrdva 4781* Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
(𝜑𝐴 ⊆ (𝐶 × 𝐷))    &   (𝜑𝐵 ⊆ (𝐶 × 𝐷))    &   ((𝜑𝑥𝐶𝑦𝐷) → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)
 
Theorembrco 4782* Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
 
Theoremopelco 4783* 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⟩ ∈ (𝐶𝐷) ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
 
Theoremcnvss 4784 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵𝐴𝐵)
 
Theoremcnveq 4785 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
(𝐴 = 𝐵𝐴 = 𝐵)
 
Theoremcnveqi 4786 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
𝐴 = 𝐵       𝐴 = 𝐵
 
Theoremcnveqd 4787 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
(𝜑𝐴 = 𝐵)       (𝜑𝐴 = 𝐵)
 
Theoremelcnv 4788* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
(𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
 
Theoremelcnv2 4789* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
(𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
 
Theoremnfcnv 4790 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥𝐴
 
Theoremopelcnvg 4791 Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
 
Theorembrcnvg 4792 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐵𝑅𝐴))
 
Theoremopelcnv 4793 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
 
Theorembrcnv 4794 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝑅𝐵𝐵𝑅𝐴)
 
Theoremcsbcnvg 4795 Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
(𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
 
Theoremcnvco 4796 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.)
(𝐴𝐵) = (𝐵𝐴)
 
Theoremcnvuni 4797* The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)
𝐴 = 𝑥𝐴 𝑥
 
Theoremdfdm3 4798* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
dom 𝐴 = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}
 
Theoremdfrn2 4799* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
 
Theoremdfrn3 4800* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
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