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Theorem List for Metamath Proof Explorer - 5601-5700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembrrelex12i 5601 Two classes that are related by a binary relation are sets. Inference form. (Contributed by BJ, 3-Oct-2022.)
Rel 𝑅       (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theorembrrelex1i 5602 The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)
Rel 𝑅       (𝐴𝑅𝐵𝐴 ∈ V)
 
Theorembrrelex2i 5603 The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐵 ∈ V)
 
Theoremnprrel12 5604 Proper classes are not related via any relation. (Contributed by AV, 29-Oct-2021.)
Rel 𝑅       (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 𝐴𝑅𝐵)
 
Theoremnprrel 5605 No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
Rel 𝑅    &    ¬ 𝐴 ∈ V        ¬ 𝐴𝑅𝐵
 
Theorem0nelrel0 5606 A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) (Revised by BJ, 14-Jul-2023.)
(Rel 𝑅 → ¬ ∅ ∈ 𝑅)
 
Theorem0nelrel 5607 A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)
(Rel 𝑅 → ∅ ∉ 𝑅)
 
Theoremfconstmpt 5608* Representation of a constant function using the mapping operation. (Note that 𝑥 cannot appear free in 𝐵.) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.)
(𝐴 × {𝐵}) = (𝑥𝐴𝐵)
 
Theoremvtoclr 5609* Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Rel 𝑅    &   ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)       ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
 
Theoremopthprc 5610 Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.)
(((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theorembrel 5611 Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝑅 ⊆ (𝐶 × 𝐷)       (𝐴𝑅𝐵 → (𝐴𝐶𝐵𝐷))
 
Theoremelxp3 5612* Membership in a Cartesian product. (Contributed by NM, 5-Mar-1995.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
 
Theoremopeliunxp 5613 Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
(⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐶𝐵))
 
Theoremxpundi 5614 Distributive law for Cartesian product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
(𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶))
 
Theoremxpundir 5615 Distributive law for Cartesian product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
 
Theoremxpiundi 5616* Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
(𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵)
 
Theoremxpiundir 5617* Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
( 𝑥𝐴 𝐵 × 𝐶) = 𝑥𝐴 (𝐵 × 𝐶)
 
Theoremiunxpconst 5618* Membership in a union of Cartesian products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
 
Theoremxpun 5619 The Cartesian product of two unions. (Contributed by NM, 12-Aug-2004.)
((𝐴𝐵) × (𝐶𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
 
Theoremelvv 5620* Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)
(𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
 
Theoremelvvv 5621* Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
(𝐴 ∈ ((V × V) × V) ↔ ∃𝑥𝑦𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
 
Theoremelvvuni 5622 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
(𝐴 ∈ (V × V) → 𝐴𝐴)
 
Theorembrinxp2 5623 Intersection with cross product binary relation. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.) Group conjuncts and avoid df-3an 1081. (Revised by Peter Mazsa, 18-Sep-2022.)
(𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶𝐴𝐷𝐵) ∧ 𝐶𝑅𝐷))
 
Theorembrinxp 5624 Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.)
((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵))
 
Theoremopelinxp 5625 Ordered pair element in an intersection with Cartesian product. (Contributed by Peter Mazsa, 21-Jul-2019.)
(⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶𝐴𝐷𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))
 
Theorempoinxp 5626 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 5627 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 5628 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 5629 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 5630 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 5631 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 5632 Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
 
Theoremfrsn 5633 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 5634 Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴))
 
Theoremelopaelxp 5635* 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 5636* 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 5637* An abstraction relation is a subset of a related Cartesian product. (Contributed by NM, 16-Jul-1995.)
{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
 
Theorembrab2a 5638* The law of concretion for a binary relation. Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 28-Apr-2015.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)}       (𝐴𝑅𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓))
 
Theoremoptocl 5639* Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)
𝐷 = (𝐵 × 𝐶)    &   (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))    &   ((𝑥𝐵𝑦𝐶) → 𝜑)       (𝐴𝐷𝜓)
 
Theorem2optocl 5640* 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 5641* 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 5642* 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 5643 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 5644 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 5645 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
 
Theoremreleqi 5646 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
𝐴 = 𝐵       (Rel 𝐴 ↔ Rel 𝐵)
 
Theoremreleqd 5647 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Rel 𝐴 ↔ Rel 𝐵))
 
Theoremnfrel 5648 Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥Rel 𝐴
 
Theoremsbcrel 5649 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 5650 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)
(𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
 
Theoremssrel 5651* 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 5195, ax-nul 5202, ax-pr 5321. (Revised by KP, 25-Oct-2021.)
(Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
 
Theoremeqrel 5652* 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 5653* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 5651 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 5654* Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
Rel 𝐴    &   (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)       𝐴𝐵
 
Theoremrelssdv 5655* Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)
(𝜑 → Rel 𝐴)    &   (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (𝜑𝐴𝐵)
 
Theoremeqrelriv 5656* 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 5657* 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 5658* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
Rel 𝐴    &   Rel 𝐵    &   (𝑥𝐴𝑦𝑥𝐵𝑦)       𝐴 = 𝐵
 
Theoremeqrelrdv 5659* 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 5660* Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑 → Rel 𝐴)    &   (𝜑 → Rel 𝐵)    &   (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)
 
Theoremeqbrrdiv 5661* Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)
Rel 𝐴    &   Rel 𝐵    &   (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)
 
Theoremeqrelrdv2 5662* A version of eqrelrdv 5659. (Contributed by Rodolfo Medina, 10-Oct-2010.)
(𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
 
Theoremssrelrel 5663* A subclass relationship determined by ordered triples. Use relrelss 6118 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 5664* Extensionality principle for ordered triples (used by 2-place operations df-oprab 7149), analogous to eqrel 5652. Use relrelss 6118 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 5665* A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
 
Theoremrel0 5666 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
Rel ∅
 
Theoremnrelv 5667 The universal class is not a relation. (Contributed by Thierry Arnoux, 23-Jan-2022.)
¬ Rel V
 
Theoremrelsng 5668 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.)
(𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
 
Theoremrelsnb 5669 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)))
 
Theoremrelsnopg 5670 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 5671 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 5672 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⟩}
 
Theoremcopsex2gb 5673* Implicit substitution inference for ordered pairs. Compare copsex2ga 5674. (Contributed by NM, 12-Mar-2014.)
(𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
 
Theoremcopsex2ga 5674* Implicit substitution inference for ordered pairs. Compare copsex2g 5376. (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 5675* 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 5676 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 5677 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 5678 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 5679 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
(Rel 𝐴 → Rel (𝐴𝐵))
 
Theoremrelin2 5680 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
(Rel 𝐵 → Rel (𝐴𝐵))
 
Theoremrelinxp 5681 Intersection with a Cartesian product is a relation. (Contributed by Peter Mazsa, 4-Mar-2019.)
Rel (𝑅 ∩ (𝐴 × 𝐵))
 
Theoremreldif 5682 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
(Rel 𝐴 → Rel (𝐴𝐵))
 
Theoremreliun 5683 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 5684 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 5685* 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 5686* 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)
 
Theoremrelopabiv 5687* A class of ordered pairs is a relation. For a version without disjoint variable condition, but a longer proof using ax-13 2383, see relopabi 5688. (Contributed by BJ, 22-Jul-2023.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       Rel 𝐴
 
Theoremrelopabi 5688 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 5195, ax-nul 5202, ax-pr 5321. (Revised by KP, 25-Oct-2021.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       Rel 𝐴
 
TheoremrelopabiALT 5689 Alternate proof of relopabi 5688 (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
 
Theoremrelopab 5690 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}
 
Theoremmptrel 5691 The maps-to notation always describes a binary relation. (Contributed by Scott Fenton, 16-Apr-2012.)
Rel (𝑥𝐴𝐵)
 
Theoremreli 5692 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 5693 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
Rel E
 
Theoremopabid2 5694* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
(Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)
 
Theoreminopab 5695* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}
 
Theoremdifopab 5696* 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 5697 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 5698 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 5699 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 5700* Distributive law for Cartesian product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐴 ≠ ∅ → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
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