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Theorem List for Metamath Proof Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelvvv 5101* Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
(𝐴 ∈ ((V × V) × V) ↔ ∃𝑥𝑦𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)

Theoremelvvuni 5102 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
(𝐴 ∈ (V × V) → 𝐴𝐴)

Theorembrinxp2 5103 Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝐶𝐵𝐷𝐴𝑅𝐵))

Theorembrinxp 5104 Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.)
((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵))

Theorempoinxp 5105 Intersection of partial order with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
(𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)

Theoremsoinxp 5106 Intersection of total order with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
(𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)

Theoremfrinxp 5107 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 5108 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 5109 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 5110 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 5111 Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Theoremfrsn 5112 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 5113 Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Theoremelopaelxp 5114* 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 5115* 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 5116* An abstraction relation is a subset of a related Cartesian product. (Contributed by NM, 16-Jul-1995.)
{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)

Theorembrab2ga 5117* The law of concretion for a binary relation. See brab2a 5091 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 5118* Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)
𝐷 = (𝐵 × 𝐶)    &   (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))    &   ((𝑥𝐵𝑦𝐶) → 𝜑)       (𝐴𝐷𝜓)

Theorem2optocl 5119* 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 5120* 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 5121* 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 5122 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 5123 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 5124 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))

Theoremreleqi 5125 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
𝐴 = 𝐵       (Rel 𝐴 ↔ Rel 𝐵)

Theoremreleqd 5126 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Rel 𝐴 ↔ Rel 𝐵))

Theoremnfrel 5127 Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥Rel 𝐴

Theoremsbcrel 5128 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 5129 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)
(𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))

Theoremssrel 5130* 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 4709, ax-nul 4717, ax-pr 4833. (Revised by KP, 25-Oct-2021.)
(Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

TheoremssrelOLD 5131* Obsolete proof of ssrel 5130 as of 25-Oct-2021. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Theoremeqrel 5132* 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 5133* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 5130 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 5134* Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
Rel 𝐴    &   (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)       𝐴𝐵

Theoremrelssdv 5135* Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)
(𝜑 → Rel 𝐴)    &   (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (𝜑𝐴𝐵)

Theoremeqrelriv 5136* 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 5137* 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 5138* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
Rel 𝐴    &   Rel 𝐵    &   (𝑥𝐴𝑦𝑥𝐵𝑦)       𝐴 = 𝐵

Theoremeqrelrdv 5139* 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 5140* Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑 → Rel 𝐴)    &   (𝜑 → Rel 𝐵)    &   (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)

Theoremeqbrrdiv 5141* Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)
Rel 𝐴    &   Rel 𝐵    &   (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)

Theoremeqrelrdv2 5142* A version of eqrelrdv 5139. (Contributed by Rodolfo Medina, 10-Oct-2010.)
(𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Theoremssrelrel 5143* A subclass relationship determined by ordered triples. Use relrelss 5576 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 5144* Extensionality principle for ordered triples (used by 2-place operations df-oprab 6553), analogous to eqrel 5132. Use relrelss 5576 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 5145* A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)

Theoremrelsn 5146 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 5147 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 5148 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 5149 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 5150 A Cartesian product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)
Rel (𝐴 × 𝐵)

Theoremxpss1 5151 Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009.)
(𝐴𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))

Theoremxpss2 5152 Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009.)
(𝐴𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵))

Theoremcopsex2gb 5153* Implicit substitution inference for ordered pairs. Compare copsex2ga 5154. (Contributed by NM, 12-Mar-2014.)
(𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))

Theoremcopsex2ga 5154* Implicit substitution inference for ordered pairs. Compare copsex2g 4884. (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 5155* 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 5156 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 5157 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 5158 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 5159 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
(Rel 𝐴 → Rel (𝐴𝐵))

Theoremrelin2 5160 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
(Rel 𝐵 → Rel (𝐴𝐵))

Theoremreldif 5161 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
(Rel 𝐴 → Rel (𝐴𝐵))

Theoremreliun 5162 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 5163 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 5164* 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 5165* 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 5166 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
Rel ∅

Theoremrelopabi 5167 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 4709, ax-nul 4717, ax-pr 4833. (Revised by KP, 25-Oct-2021.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       Rel 𝐴

TheoremrelopabiALT 5168 Alternate proof of relopabi 5167. (Contributed by Mario Carneiro, 21-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       Rel 𝐴

Theoremrelopab 5169 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 5170 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
Rel (𝑥𝐴𝐵)

Theoremreli 5171 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 5172 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
Rel E

Theoremopabid2 5173* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
(Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)

Theoreminopab 5174* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}

Theoremdifopab 5175* 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 5176 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 5177 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 5178 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 5179* Distributive law for Cartesian product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐴 ≠ ∅ → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))

Theoremxpriindi 5180* 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 5181* Membership in a union of Cartesian products. Analogue of elxp 5055 for nonconstant 𝐵(𝑥). (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))

Theoremopeliunxp2 5182* Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 14-Feb-2015.)
(𝑥 = 𝐶𝐵 = 𝐸)       (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))

Theoremraliunxp 5183* Write a double restricted quantification as one universal quantifier. In this version of ralxp 5185, 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 5184* Write a double restricted quantification as one universal quantifier. In this version of rexxp 5186, 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 5185* 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 5186* 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 5187* 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 5188* 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 5189* Disjoint union is a subset of a Cartesian product. (Contributed by Stefan O'Rear, 21-Nov-2014.)
𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)

Theoremralxpf 5190* Version of ralxp 5185 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 5191* Version of rexxp 5186 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 5192* 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 5193* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2729. (Contributed by NM, 24-Feb-2014.)
Rel 𝐴    &   (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝜓))       (𝜑𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})

Theoremrelop 5194* 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 5195 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 5196 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
𝐵 ∈ V       (𝐴 I 𝐵𝐴 = 𝐵)

Theoremididg 5197 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝑉𝐴 I 𝐴)

Theoremissetid 5198 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 5199 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremcoss2 5200 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

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