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Theorem List for Metamath Proof Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrinxp 5001 Intersection of well-founded relation with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
(𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)
 
Theoremseinxp 5002 Intersection of set-like relation with Cartesian product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
(𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)
 
Theoremweinxp 5003 Intersection of well-ordering with Cartesian product of its field. (Contributed by NM, 9-Mar-1997.) (Revised by Mario Carneiro, 10-Jul-2014.)
(𝑅 We 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) We 𝐴)
 
Theoremposn 5004 Partial ordering of a singleton. (Contributed by NM, 27-Apr-2009.) (Revised by Mario Carneiro, 23-Apr-2015.)
(Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
 
Theoremsosn 5005 Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
 
Theoremfrsn 5006 Founded relation on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 23-Apr-2015.)
(Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
 
Theoremwesn 5007 Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴))
 
Theoremelopaelxp 5008* Membership in an ordered pair class builder implies membership in a Cartesian product. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
(𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))
 
Theorembropaex12 5009* Two classes related by an ordered pair class builder are sets. (Contributed by AV, 21-Jan-2020.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}       (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theoremopabssxp 5010* An abstraction relation is a subset of a related Cartesian product. (Contributed by NM, 16-Jul-1995.)
{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
 
Theorembrab2ga 5011* The law of concretion for a binary relation. See brab2a 4985 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)}       (𝐴𝑅𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓))
 
Theoremoptocl 5012* Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)
𝐷 = (𝐵 × 𝐶)    &   (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))    &   ((𝑥𝐵𝑦𝐶) → 𝜑)       (𝐴𝐷𝜓)
 
Theorem2optocl 5013* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
𝑅 = (𝐶 × 𝐷)    &   (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))    &   (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓𝜒))    &   (((𝑥𝐶𝑦𝐷) ∧ (𝑧𝐶𝑤𝐷)) → 𝜑)       ((𝐴𝑅𝐵𝑅) → 𝜒)
 
Theorem3optocl 5014* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
𝑅 = (𝐷 × 𝐹)    &   (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))    &   (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓𝜒))    &   (⟨𝑣, 𝑢⟩ = 𝐶 → (𝜒𝜃))    &   (((𝑥𝐷𝑦𝐹) ∧ (𝑧𝐷𝑤𝐹) ∧ (𝑣𝐷𝑢𝐹)) → 𝜑)       ((𝐴𝑅𝐵𝑅𝐶𝑅) → 𝜃)
 
Theoremopbrop 5015* Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)
(((𝑧 = 𝐴𝑤 = 𝐵) ∧ (𝑣 = 𝐶𝑢 = 𝐷)) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}       (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ 𝜓))
 
Theorem0xp 5016 The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)
(∅ × 𝐴) = ∅
 
Theoremcsbxp 5017 Distribute proper substitution through the Cartesian product of two classes. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 23-Aug-2018.)
𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶)
 
Theoremreleq 5018 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
 
Theoremreleqi 5019 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
𝐴 = 𝐵       (Rel 𝐴 ↔ Rel 𝐵)
 
Theoremreleqd 5020 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Rel 𝐴 ↔ Rel 𝐵))
 
Theoremnfrel 5021 Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥Rel 𝐴
 
Theoremsbcrel 5022 Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel 𝐴 / 𝑥𝑅))
 
Theoremrelss 5023 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)
(𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
 
Theoremssrel 5024* 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.)
(Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
 
Theoremeqrel 5025* Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)
((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
 
Theoremssrel2 5026* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 5024 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 5027* Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
Rel 𝐴    &   (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)       𝐴𝐵
 
Theoremrelssdv 5028* Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)
(𝜑 → Rel 𝐴)    &   (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (𝜑𝐴𝐵)
 
Theoremeqrelriv 5029* 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 5030* 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 5031* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
Rel 𝐴    &   Rel 𝐵    &   (𝑥𝐴𝑦𝑥𝐵𝑦)       𝐴 = 𝐵
 
Theoremeqrelrdv 5032* 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 5033* Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑 → Rel 𝐴)    &   (𝜑 → Rel 𝐵)    &   (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)
 
Theoremeqbrrdiv 5034* Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)
Rel 𝐴    &   Rel 𝐵    &   (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)
 
Theoremeqrelrdv2 5035* A version of eqrelrdv 5032. (Contributed by Rodolfo Medina, 10-Oct-2010.)
(𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
 
Theoremssrelrel 5036* A subclass relationship determined by ordered triples. Use relrelss 5466 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 5037* Extensionality principle for ordered triples (used by 2-place operations df-oprab 6430), analogous to eqrel 5025. Use relrelss 5466 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 5038* A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
 
Theoremrelsn 5039 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 5040 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 5041 Subset theorem for Cartesian 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 5042 A Cartesian 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 5043 A Cartesian product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)
Rel (𝐴 × 𝐵)
 
Theoremxpss1 5044 Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009.)
(𝐴𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))
 
Theoremxpss2 5045 Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009.)
(𝐴𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵))
 
Theoremcopsex2gb 5046* Implicit substitution inference for ordered pairs. Compare copsex2ga 5047. (Contributed by NM, 12-Mar-2014.)
(𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
 
Theoremcopsex2ga 5047* Implicit substitution inference for ordered pairs. Compare copsex2g 4782. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
(𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       (𝐴 ∈ (𝑉 × 𝑊) → (𝜑 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
 
Theoremelopaba 5048* Membership in an ordered pair class builder. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
 
Theoremxpsspw 5049 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 (𝐴𝐵)
 
Theoremunixpss 5050 The double class union of a Cartesian product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)
(𝐴 × 𝐵) ⊆ (𝐴𝐵)
 
Theoremrelun 5051 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 5052 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
(Rel 𝐴 → Rel (𝐴𝐵))
 
Theoremrelin2 5053 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
(Rel 𝐵 → Rel (𝐴𝐵))
 
Theoremreldif 5054 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
(Rel 𝐴 → Rel (𝐴𝐵))
 
Theoremreliun 5055 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 5056 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 5057* 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 5058* 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 5059 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
Rel ∅
 
Theoremrelopabi 5060 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       Rel 𝐴
 
Theoremrelopab 5061 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 5062 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
Rel (𝑥𝐴𝐵)
 
Theoremreli 5063 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 5064 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
Rel E
 
Theoremopabid2 5065* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
(Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)
 
Theoreminopab 5066* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}
 
Theoremdifopab 5067* 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 5068 The 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)) = ((𝐴𝐶) × (𝐵𝐷))
 
Theoremxpindi 5069 Distributive law for Cartesian product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
(𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))
 
Theoremxpindir 5070 Distributive law for Cartesian product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))
 
Theoremxpiindi 5071* Distributive law for Cartesian product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐴 ≠ ∅ → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
 
Theoremxpriindi 5072* Distributive law for Cartesian product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵))
 
Theoremeliunxp 5073* Membership in a union of Cartesian products. Analogue of elxp 4949 for nonconstant 𝐵(𝑥). (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
 
Theoremopeliunxp2 5074* Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 14-Feb-2015.)
(𝑥 = 𝐶𝐵 = 𝐸)       (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
 
Theoremraliunxp 5075* Write a double restricted quantification as one universal quantifier. In this version of ralxp 5077, 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 5076* Write a double restricted quantification as one universal quantifier. In this version of rexxp 5078, 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 5077* 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⟩ → (𝜑𝜓))       (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
 
Theoremrexxp 5078* 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⟩ → (𝜑𝜓))       (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
 
Theoremexopxfr 5079* 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⟩ → (𝜑𝜓))       (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦𝑧𝜓)
 
Theoremexopxfr2 5080* Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.)
(𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))       (Rel 𝐴 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓)))
 
Theoremdjussxp 5081* Disjoint union is a subset of a Cartesian product. (Contributed by Stefan O'Rear, 21-Nov-2014.)
𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)
 
Theoremralxpf 5082* Version of ralxp 5077 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 5083* Version of rexxp 5078 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 5084* Indexed union on a Cartesian product 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 5085* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2633. (Contributed by NM, 24-Feb-2014.)
Rel 𝐴    &   (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝜓))       (𝜑𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
 
Theoremrelop 5086* 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 5087 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 5088 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
𝐵 ∈ V       (𝐴 I 𝐵𝐴 = 𝐵)
 
Theoremididg 5089 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝑉𝐴 I 𝐴)
 
Theoremissetid 5090 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 5091 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremcoss2 5092 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremcoeq1 5093 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremcoeq2 5094 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremcoeq1i 5095 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremcoeq2i 5096 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremcoeq1d 5097 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremcoeq2d 5098 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremcoeq12i 5099 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)
 
Theoremcoeq12d 5100 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))
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