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Theorem List for Metamath Proof Explorer - 36701-36800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremidresssidinxp 36701 Condition for the identity restriction to be a subclass of identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018.)
(𝐴𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))
 
Theoremidreseqidinxp 36702 Condition for the identity restriction to be equal to the identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018.)
(𝐴𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴))
 
Theoremextid 36703 Property of identity relation, see also extep 36675, extssr 36903 and the comment of df-ssr 36892. (Contributed by Peter Mazsa, 5-Jul-2019.)
(𝐴𝑉 → ([𝐴] I = [𝐵] I ↔ 𝐴 = 𝐵))
 
Theoreminxpss 36704* Two ways to say that an intersection with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019.)
((𝑅 ∩ (𝐴 × 𝐵)) ⊆ 𝑆 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
 
Theoremidinxpss 36705* Two ways to say that an intersection of the identity relation with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019.)
(( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
 
Theoremref5 36706* Two ways to say that an intersection of the identity relation with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 12-Dec-2023.)
(( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥 ∈ (𝐴𝐵)𝑥𝑅𝑥)
 
Theoreminxpss3 36707* Two ways to say that an intersection with a Cartesian product is a subclass (see also inxpss 36704). (Contributed by Peter Mazsa, 8-Mar-2019.)
(∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
 
Theoreminxpss2 36708* Two ways to say that intersections with Cartesian products are in a subclass relation. (Contributed by Peter Mazsa, 8-Mar-2019.)
((𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑆 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
 
Theoreminxpssidinxp 36709* Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 36708. (Contributed by Peter Mazsa, 4-Jul-2019.)
((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))
 
Theoremidinxpssinxp 36710* Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 36708. (Contributed by Peter Mazsa, 6-Mar-2019.)
(( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
 
Theoremidinxpssinxp2 36711* Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 4-Mar-2019.) (Proof modification is discouraged.)
(( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
 
Theoremidinxpssinxp3 36712 Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 16-Mar-2019.) (Proof modification is discouraged.)
(( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ( I ↾ 𝐴) ⊆ 𝑅)
 
Theoremidinxpssinxp4 36713* Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (see also idinxpssinxp2 36711). (Contributed by Peter Mazsa, 8-Mar-2019.)
(∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
 
Theoremrelcnveq3 36714* Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.)
(Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
 
Theoremrelcnveq 36715 Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.)
(Rel 𝑅 → (𝑅𝑅𝑅 = 𝑅))
 
Theoremrelcnveq2 36716* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)
(Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
 
Theoremrelcnveq4 36717* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)
(Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
 
Theoremqsresid 36718 Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.)
(𝐴 / (𝑅𝐴)) = (𝐴 / 𝑅)
 
Theoremn0elqs 36719 Two ways of expressing that the empty set is not an element of a quotient set. (Contributed by Peter Mazsa, 5-Dec-2019.)
(¬ ∅ ∈ (𝐴 / 𝑅) ↔ 𝐴 ⊆ dom 𝑅)
 
Theoremn0elqs2 36720 Two ways of expressing that the empty set is not an element of a quotient set. (Contributed by Peter Mazsa, 25-Jul-2021.)
(¬ ∅ ∈ (𝐴 / 𝑅) ↔ dom (𝑅𝐴) = 𝐴)
 
Theoremecex2 36721 Condition for a coset to be a set. (Contributed by Peter Mazsa, 4-May-2019.)
((𝑅𝐴) ∈ 𝑉 → (𝐵𝐴 → [𝐵]𝑅 ∈ V))
 
TheoremuniqsALTV 36722 The union of a quotient set, like uniqs 8674 but with a weaker antecedent: only the restricion of 𝑅 by 𝐴 needs to be a set, not 𝑅 itself, see e.g. cnvepima 36730. (Contributed by Peter Mazsa, 20-Jun-2019.)
((𝑅𝐴) ∈ 𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
 
TheoremimaexALTV 36723 Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7844) with weakened antecedent: only the restricion of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 36729. (Contributed by Peter Mazsa, 22-Feb-2023.) (Proof modification is discouraged.)
((𝐴𝑉 ∨ ((𝐴𝐵) ∈ 𝑊𝐵𝑋)) → (𝐴𝐵) ∈ V)
 
TheoremecexALTV 36724 Existence of a coset, like ecexg 8610 but with a weaker antecedent: only the restricion of 𝑅 by the singleton of 𝐴 needs to be a set, not 𝑅 itself, see e.g. eccnvepex 36728. (Contributed by Peter Mazsa, 22-Feb-2023.)
((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V)
 
Theoremrnresequniqs 36725 The range of a restriction is equal to the union of the quotient set. (Contributed by Peter Mazsa, 19-May-2018.)
((𝑅𝐴) ∈ 𝑉 → ran (𝑅𝐴) = (𝐴 / 𝑅))
 
Theoremn0el2 36726 Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 31-Jan-2018.)
(¬ ∅ ∈ 𝐴 ↔ dom ( E ↾ 𝐴) = 𝐴)
 
Theoremcnvepresex 36727 Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.)
(𝐴𝑉 → ( E ↾ 𝐴) ∈ V)
 
Theoremeccnvepex 36728 The converse epsilon coset exists. (Contributed by Peter Mazsa, 22-Mar-2023.)
[𝐴] E ∈ V
 
Theoremcnvepimaex 36729 The image of converse epsilon exists, proof via imaexALTV 36723 (see also cnvepima 36730 and uniexg 7669 for alternate way). (Contributed by Peter Mazsa, 22-Mar-2023.)
(𝐴𝑉 → ( E “ 𝐴) ∈ V)
 
Theoremcnvepima 36730 The image of converse epsilon. (Contributed by Peter Mazsa, 22-Mar-2023.)
(𝐴𝑉 → ( E “ 𝐴) = 𝐴)
 
Theoreminex3 36731 Sufficient condition for the intersection relation to be a set. (Contributed by Peter Mazsa, 24-Nov-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theoreminxpex 36732 Sufficient condition for an intersection with a Cartesian product to be a set. (Contributed by Peter Mazsa, 10-May-2019.)
((𝑅𝑊 ∨ (𝐴𝑈𝐵𝑉)) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
 
Theoremeqres 36733 Converting a class constant definition by restriction (like df-ers 37057 or df-parts 37159) into a binary relation. (Contributed by Peter Mazsa, 1-Oct-2018.)
𝑅 = (𝑆𝐶)       (𝐵𝑉 → (𝐴𝑅𝐵 ↔ (𝐴𝐶𝐴𝑆𝐵)))
 
Theorembrrabga 36734* The law of concretion for operation class abstraction. (Contributed by Peter Mazsa, 24-Oct-2022.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))    &   𝑅 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵𝑅𝐶𝜓))
 
Theorembrcnvrabga 36735* The law of concretion for the converse of operation class abstraction. (Contributed by Peter Mazsa, 25-Oct-2022.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))    &   𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑅𝐵, 𝐶⟩ ↔ 𝜓))
 
Theoremopideq 36736 Equality conditions for ordered pairs 𝐴, 𝐴 and 𝐵, 𝐵. (Contributed by Peter Mazsa, 22-Jul-2019.) (Revised by Thierry Arnoux, 16-Feb-2022.)
(𝐴𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ 𝐴 = 𝐵))
 
Theoremiss2 36737 A subclass of the identity relation is the intersection of identity relation with Cartesian product of the domain and range of the class. (Contributed by Peter Mazsa, 22-Jul-2019.)
(𝐴 ⊆ I ↔ 𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)))
 
Theoremeldmcnv 36738* Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.)
(𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
 
Theoremdfrel5 36739 Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 6-Nov-2018.)
(Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
 
Theoremdfrel6 36740 Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.)
(Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
 
Theoremcnvresrn 36741 Converse restricted to range is converse. (Contributed by Peter Mazsa, 3-Sep-2021.)
(𝑅 ↾ ran 𝑅) = 𝑅
 
Theoremrelssinxpdmrn 36742 Subset of restriction, special case. (Contributed by Peter Mazsa, 10-Apr-2023.)
(Rel 𝑅 → (𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅𝑆))
 
Theoremcnvref4 36743 Two ways to say that a relation is a subclass. (Contributed by Peter Mazsa, 11-Apr-2023.)
(Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅𝑆))
 
Theoremcnvref5 36744* Two ways to say that a relation is a subclass of the identity relation. (Contributed by Peter Mazsa, 26-Jun-2019.)
(Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑥 = 𝑦)))
 
Theoremecin0 36745* Two ways of saying that the coset of 𝐴 and the coset of 𝐵 have no elements in common. (Contributed by Peter Mazsa, 1-Dec-2018.)
((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
 
Theoremecinn0 36746* Two ways of saying that the coset of 𝐴 and the coset of 𝐵 have some elements in common. (Contributed by Peter Mazsa, 23-Jan-2019.)
((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
 
Theoremineleq 36747* Equivalence of restricted universal quantifications. (Contributed by Peter Mazsa, 29-May-2018.)
(∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
 
Theoreminecmo 36748* Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 29-May-2018.)
(𝑥 = 𝑦𝐵 = 𝐶)       (Rel 𝑅 → (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥𝐴 𝐵𝑅𝑧))
 
Theoreminecmo2 36749* Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 29-May-2018.) (Revised by Peter Mazsa, 2-Sep-2021.)
((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢𝐴 𝑢𝑅𝑥 ∧ Rel 𝑅))
 
Theoremineccnvmo 36750* Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.)
(∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
 
Theoremalrmomorn 36751 Equivalence of an "at most one" and an "at most one" restricted to the range inside a universal quantification. (Contributed by Peter Mazsa, 3-Sep-2021.)
(∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦)
 
Theoremalrmomodm 36752* Equivalence of an "at most one" and an "at most one" restricted to the domain inside a universal quantification. (Contributed by Peter Mazsa, 5-Sep-2021.)
(Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥))
 
Theoremineccnvmo2 36753* Equivalence of a double universal quantification restricted to the range and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 4-Sep-2021.)
(∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)
 
Theoreminecmo3 36754* Equivalence of a double universal quantification restricted to the domain and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 5-Sep-2021.)
((∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅))
 
Theoremmoeu2 36755 Uniqueness is equivalent to non-existence or unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by Peter Mazsa, 19-Nov-2024.)
(∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑))
 
Theoremmopickr 36756 "At most one" picks a variable value, eliminating an existential quantifier. The proof begins with references *2.21 (pm2.21 123) and *14.26 (eupickbi 2636) from [WhiteheadRussell] p. 104 and p. 183. (Contributed by Peter Mazsa, 18-Nov-2024.) (Proof modification is discouraged.)
((∃*𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜓𝜑))
 
Theoremmoantr 36757 Sufficient condition for transitivity of conjunctions inside existential quantifiers. (Contributed by Peter Mazsa, 2-Oct-2018.)
(∃*𝑥𝜓 → ((∃𝑥(𝜑𝜓) ∧ ∃𝑥(𝜓𝜒)) → ∃𝑥(𝜑𝜒)))
 
Theorembrabidgaw 36758* The law of concretion for a binary relation. Special case of brabga 5489. Version of brabidga 36759 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Peter Mazsa, 24-Nov-2018.) (Revised by Gino Giotto, 2-Apr-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       (𝑥𝑅𝑦𝜑)
 
Theorembrabidga 36759 The law of concretion for a binary relation. Special case of brabga 5489. Usage of this theorem is discouraged because it depends on ax-13 2370, see brabidgaw 36758 for a weaker version that does not require it. (Contributed by Peter Mazsa, 24-Nov-2018.) (New usage is discouraged.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       (𝑥𝑅𝑦𝜑)
 
Theoreminxp2 36760* Intersection with a Cartesian product. (Contributed by Peter Mazsa, 18-Jul-2019.)
(𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦)}
 
Theoremopabf 36761 A class abstraction of a collection of ordered pairs with a negated wff is the empty set. (Contributed by Peter Mazsa, 21-Oct-2019.) (Proof shortened by Thierry Arnoux, 18-Feb-2022.)
¬ 𝜑       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅
 
Theoremec0 36762 The empty-coset of a class is the empty set. (Contributed by Peter Mazsa, 19-May-2019.)
[𝐴]∅ = ∅
 
Theorem0qs 36763 Quotient set with the empty set. (Contributed by Peter Mazsa, 14-Sep-2019.)
(∅ / 𝑅) = ∅
 
Theorembrcnvin 36764 Intersection with a converse, binary relation. (Contributed by Peter Mazsa, 24-Mar-2024.)
((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐵𝑆𝐴)))
 
21.22.3  Range Cartesian product
 
Definitiondf-xrn 36765 Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 36766 and brxrn 36768. This is Scott Fenton's df-txp 34371 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 34371. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
 
Theoremxrnss3v 36766 A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 34395 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 34395. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) ⊆ (V × (V × V))
 
Theoremxrnrel 36767 A range Cartesian product is a relation. This is Scott Fenton's txprel 34396 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 34396. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel (𝐴𝐵)
 
Theorembrxrn 36768 Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 36766, this characterizes elementhood in a range cross. (Contributed by Peter Mazsa, 27-Jun-2021.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵𝐴𝑆𝐶)))
 
Theorembrxrn2 36769* A characterization of the range Cartesian product. (Contributed by Peter Mazsa, 14-Oct-2020.)
(𝐴𝑉 → (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
 
Theoremdfxrn2 36770* Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022.)
(𝑅𝑆) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}
 
Theoremxrneq1 36771 Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremxrneq1i 36772 Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremxrneq1d 36773 Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremxrneq2 36774 Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremxrneq2i 36775 Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremxrneq2d 36776 Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremxrneq12 36777 Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
 
Theoremxrneq12i 36778 Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)
 
Theoremxrneq12d 36779 Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 
Theoremelecxrn 36780* Elementhood in the (𝑅𝑆)-coset of 𝐴. (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
(𝐴𝑉 → (𝐵 ∈ [𝐴](𝑅𝑆) ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
 
Theoremecxrn 36781* The (𝑅𝑆)-coset of 𝐴. (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
(𝐴𝑉 → [𝐴](𝑅𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)})
 
Theoremdisjressuc2 36782* Double restricted quantification over the union of a set and its singleton. (Contributed by Peter Mazsa, 22-Aug-2023.)
(𝐴𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
 
Theoremdisjecxrn 36783 Two ways of saying that (𝑅𝑆)-cosets are disjoint. (Contributed by Peter Mazsa, 19-Jun-2020.) (Revised by Peter Mazsa, 21-Aug-2023.)
((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅)))
 
Theoremdisjecxrncnvep 36784 Two ways of saying that cosets are disjoint, special case of disjecxrn 36783. (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, 25-Aug-2023.)
((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅 E ) ∩ [𝐵](𝑅 E )) = ∅ ↔ ((𝐴𝐵) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅)))
 
Theoremdisjsuc2 36785* Double restricted quantification over the union of a set and its singleton. (Contributed by Peter Mazsa, 22-Aug-2023.)
(𝐴𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
 
Theoremxrninxp 36786* Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 7-Apr-2020.)
((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)⟨𝑦, 𝑧⟩))}
 
Theoremxrninxp2 36787* Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020.)
((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
 
Theoremxrninxpex 36788 Sufficient condition for the intersection of a range Cartesian product with a Cartesian product to be a set. (Contributed by Peter Mazsa, 12-Apr-2020.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
 
Theoreminxpxrn 36789 Two ways to express the intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 10-Apr-2020.)
((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶))) = ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))
 
Theorembr1cnvxrn2 36790* The converse of a binary relation over a range Cartesian product. (Contributed by Peter Mazsa, 11-Jul-2021.)
(𝐵𝑉 → (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝐵𝑅𝑦𝐵𝑆𝑧)))
 
Theoremelec1cnvxrn2 36791* Elementhood in the converse range Cartesian product coset of 𝐴. (Contributed by Peter Mazsa, 11-Jul-2021.)
(𝐵𝑉 → (𝐵 ∈ [𝐴](𝑅𝑆) ↔ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝐵𝑅𝑦𝐵𝑆𝑧)))
 
Theoremrnxrn 36792* Range of the range Cartesian product of classes. (Contributed by Peter Mazsa, 1-Jun-2020.)
ran (𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)}
 
Theoremrnxrnres 36793* Range of a range Cartesian product with a restricted relation. (Contributed by Peter Mazsa, 5-Dec-2021.)
ran (𝑅 ⋉ (𝑆𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦)}
 
Theoremrnxrncnvepres 36794* Range of a range Cartesian product with a restriction of the converse epsilon relation. (Contributed by Peter Mazsa, 6-Dec-2021.)
ran (𝑅 ⋉ ( E ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑦𝑢𝑢𝑅𝑥)}
 
Theoremrnxrnidres 36795* Range of a range Cartesian product with a restriction of the identity relation. (Contributed by Peter Mazsa, 6-Dec-2021.)
ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥)}
 
Theoremxrnres 36796 Two ways to express restriction of range Cartesian product, see also xrnres2 36797, xrnres3 36798. (Contributed by Peter Mazsa, 5-Jun-2021.)
((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ 𝑆)
 
Theoremxrnres2 36797 Two ways to express restriction of range Cartesian product, see also xrnres 36796, xrnres3 36798. (Contributed by Peter Mazsa, 6-Sep-2021.)
((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))
 
Theoremxrnres3 36798 Two ways to express restriction of range Cartesian product, see also xrnres 36796, xrnres2 36797. (Contributed by Peter Mazsa, 28-Mar-2020.)
((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))
 
Theoremxrnres4 36799 Two ways to express restriction of range Cartesian product. (Contributed by Peter Mazsa, 29-Dec-2020.)
((𝑅𝑆) ↾ 𝐴) = ((𝑅𝑆) ∩ (𝐴 × (ran (𝑅𝐴) × ran (𝑆𝐴))))
 
Theoremxrnresex 36800 Sufficient condition for a restricted range Cartesian product to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 7-Sep-2021.)
((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆𝐴)) ∈ V)
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