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Theorem List for Metamath Proof Explorer - 35601-35700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremideq2 35601 For sets, the identity binary relation is the same as equality. (Contributed by Peter Mazsa, 24-Jun-2020.) (Revised by Peter Mazsa, 18-Dec-2021.)
(𝐴𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))

Theoremidresssidinxp 35602 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 35603 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 35604 Property of identity relation, see also extep 35576, extssr 35785 and the comment of df-ssr 35774. (Contributed by Peter Mazsa, 5-Jul-2019.)
(𝐴𝑉 → ([𝐴] I = [𝐵] I ↔ 𝐴 = 𝐵))

Theoreminxpss 35605* 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 35606* 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))

Theoreminxpss3 35607* Two ways to say that an intersection with a Cartesian product is a subclass (see also inxpss 35605). (Contributed by Peter Mazsa, 8-Mar-2019.)
(∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))

Theoreminxpss2 35608* 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 35609* Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 35608. (Contributed by Peter Mazsa, 4-Jul-2019.)
((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))

Theoremidinxpssinxp 35610* Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 35608. (Contributed by Peter Mazsa, 6-Mar-2019.)
(( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))

Theoremidinxpssinxp2 35611* 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 35612 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 35613* Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (see also idinxpssinxp2 35611). (Contributed by Peter Mazsa, 8-Mar-2019.)
(∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)

Theoremrelcnveq3 35614* Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.)
(Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

Theoremrelcnveq 35615 Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.)
(Rel 𝑅 → (𝑅𝑅𝑅 = 𝑅))

Theoremrelcnveq2 35616* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)
(Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

Theoremrelcnveq4 35617* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)
(Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

Theoremqsresid 35618 Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.)
(𝐴 / (𝑅𝐴)) = (𝐴 / 𝑅)

Theoremn0elqs 35619 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 35620 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 35621 Condition for a coset to be a set. (Contributed by Peter Mazsa, 4-May-2019.)
((𝑅𝐴) ∈ 𝑉 → (𝐵𝐴 → [𝐵]𝑅 ∈ V))

TheoremuniqsALTV 35622 The union of a quotient set, like uniqs 8335 but with a weaker antecedent: only the restricion of 𝑅 by 𝐴 needs to be a set, not 𝑅 itself, see e.g. cnvepima 35630. (Contributed by Peter Mazsa, 20-Jun-2019.)
((𝑅𝐴) ∈ 𝑉 (𝐴 / 𝑅) = (𝑅𝐴))

TheoremimaexALTV 35623 Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7598) with weakened antecedent: only the restricion of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 35629. (Contributed by Peter Mazsa, 22-Feb-2023.) (Proof modification is discouraged.)
((𝐴𝑉 ∨ ((𝐴𝐵) ∈ 𝑊𝐵𝑋)) → (𝐴𝐵) ∈ V)

TheoremecexALTV 35624 Existence of a coset, like ecexg 8271 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 35628. (Contributed by Peter Mazsa, 22-Feb-2023.)
((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V)

Theoremrnresequniqs 35625 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 35626 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 35627 Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.)
(𝐴𝑉 → ( E ↾ 𝐴) ∈ V)

Theoremeccnvepex 35628 The converse epsilon coset exists. (Contributed by Peter Mazsa, 22-Mar-2023.)
[𝐴] E ∈ V

Theoremcnvepimaex 35629 The image of converse epsilon exists, proof via imaexALTV 35623 (see also cnvepima 35630 and uniexg 7444 for alternate way). (Contributed by Peter Mazsa, 22-Mar-2023.)
(𝐴𝑉 → ( E “ 𝐴) ∈ V)

Theoremcnvepima 35630 The image of converse epsilon. (Contributed by Peter Mazsa, 22-Mar-2023.)
(𝐴𝑉 → ( E “ 𝐴) = 𝐴)

Theoreminex3 35631 Sufficient condition for the intersection relation to be a set. (Contributed by Peter Mazsa, 24-Nov-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Theoreminxpex 35632 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 35633 Converting a class constant definition by restriction (like df-ers 35933 or ~? df-parts ) into a binary relation. (Contributed by Peter Mazsa, 1-Oct-2018.)
𝑅 = (𝑆𝐶)       (𝐵𝑉 → (𝐴𝑅𝐵 ↔ (𝐴𝐶𝐴𝑆𝐵)))

Theorembrrabga 35634* 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 35635* 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 35636 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 35637 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 35638* Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.)
(𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

Theoremdfrel5 35639 Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 6-Nov-2018.)
(Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)

Theoremdfrel6 35640 Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.)
(Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)

Theoremcnvresrn 35641 Converse restricted to range is converse. (Contributed by Peter Mazsa, 3-Sep-2021.)
(𝑅 ↾ ran 𝑅) = 𝑅

Theoremecin0 35642* 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 35643* 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 35644* Equivalence of restricted universal quantifications. (Contributed by Peter Mazsa, 29-May-2018.)
(∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))

Theoreminecmo 35645* 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 35646* 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 35647* 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 35648 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 35649* 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 35650* 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 35651* 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))

Theoremmoantr 35652 Sufficient condition for transitivity of conjunctions inside existential quantifiers. (Contributed by Peter Mazsa, 2-Oct-2018.)
(∃*𝑥𝜓 → ((∃𝑥(𝜑𝜓) ∧ ∃𝑥(𝜓𝜒)) → ∃𝑥(𝜑𝜒)))

Theorembrabidgaw 35653* The law of concretion for a binary relation. Special case of brabga 5397. Version of brabidga 35654 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by Peter Mazsa, 24-Nov-2018.) (Revised by Gino Giotto, 2-Apr-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       (𝑥𝑅𝑦𝜑)

Theorembrabidga 35654 The law of concretion for a binary relation. Special case of brabga 5397. Usage of this theorem is discouraged because it depends on ax-13 2390, see brabidgaw 35653 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 35655* Intersection with a Cartesian product. (Contributed by Peter Mazsa, 18-Jul-2019.)
(𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦)}

Theoremopabf 35656 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 35657 The empty-coset of a class is the empty set. (Contributed by Peter Mazsa, 19-May-2019.)
[𝐴]∅ = ∅

Theorem0qs 35658 Quotient set with the empty set. (Contributed by Peter Mazsa, 14-Sep-2019.)
(∅ / 𝑅) = ∅

20.22.3  Range Cartesian product

Definitiondf-xrn 35659 Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 35660 and brxrn 35662. This is Scott Fenton's df-txp 33323 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 33323. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))

Theoremxrnss3v 35660 A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 33347 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 33347. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) ⊆ (V × (V × V))

Theoremxrnrel 35661 A range Cartesian product is a relation. This is Scott Fenton's txprel 33348 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 33348. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel (𝐴𝐵)

Theorembrxrn 35662 Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 35660, this characterizes elementhood in a range cross. (Contributed by Peter Mazsa, 27-Jun-2021.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵𝐴𝑆𝐶)))

Theorembrxrn2 35663* A characterization of the range Cartesian product. (Contributed by Peter Mazsa, 14-Oct-2020.)
(𝐴𝑉 → (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))

Theoremdfxrn2 35664* Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022.)
(𝑅𝑆) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}

Theoremxrneq1 35665 Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremxrneq1i 35666 Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremxrneq1d 35667 Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremxrneq2 35668 Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremxrneq2i 35669 Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremxrneq2d 35670 Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremxrneq12 35671 Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremxrneq12i 35672 Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremxrneq12d 35673 Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremelecxrn 35674* 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 35675* 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)})

Theoremxrninxp 35676* 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 35677* 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 35678 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 35679 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 35680* 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 35681* 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 35682* Range of the range Cartesian product of classes. (Contributed by Peter Mazsa, 1-Jun-2020.)
ran (𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)}

Theoremrnxrnres 35683* 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 35684* 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 35685* 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 35686 Two ways to express restriction of range Cartesian product, see also xrnres2 35687, xrnres3 35688. (Contributed by Peter Mazsa, 5-Jun-2021.)
((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ 𝑆)

Theoremxrnres2 35687 Two ways to express restriction of range Cartesian product, see also xrnres 35686, xrnres3 35688. (Contributed by Peter Mazsa, 6-Sep-2021.)
((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))

Theoremxrnres3 35688 Two ways to express restriction of range Cartesian product, see also xrnres 35686, xrnres2 35687. (Contributed by Peter Mazsa, 28-Mar-2020.)
((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))

Theoremxrnres4 35689 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 35690 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)

Theoremxrnidresex 35691 Sufficient condition for a range Cartesian product with restricted identity to be a set. (Contributed by Peter Mazsa, 31-Dec-2021.)
((𝐴𝑉𝑅𝑊) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V)

Theoremxrncnvepresex 35692 Sufficient condition for a range Cartesian product with restricted converse epsilon to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 23-Sep-2021.)
((𝐴𝑉𝑅𝑊) → (𝑅 ⋉ ( E ↾ 𝐴)) ∈ V)

Theorembrin2 35693 Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩))

Theorembrin3 35694 Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.) (Avoid depending on this detail.)
((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆){{𝐵}}))

20.22.4  Cosets by ` R `

Definitiondf-coss 35695* Define the class of cosets by 𝑅: 𝑥 and 𝑦 are cosets by 𝑅 iff there exists a set 𝑢 such that both 𝑢𝑅𝑥 and 𝑢𝑅𝑦 hold, i.e., both 𝑥 and 𝑦 are are elements of the 𝑅 -coset of 𝑢 (see dfcoss2 35697 and the comment of dfec2 8270). 𝑅 is usually a relation.

This concept simplifies theorems relating partition and equivalence: the left side of these theorems relate to 𝑅, the right side relate to 𝑅 (see e.g. ~? pet ). Without the definition of 𝑅 we should have to relate the right side of these theorems to a composition of a converse (cf. dfcoss3 35698) or to the range of a range Cartesian product of classes (cf. dfcoss4 35699), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 35697. Technically, we can define it via composition (dfcoss3 35698) or as the range of a range Cartesian product (dfcoss4 35699), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions (df-funsALTV 35950, df-funALTV 35951) and disjoints (dfdisjs 35977, dfdisjs2 35978, df-disjALTV 35974, dfdisjALTV2 35983) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.)

𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}

Definitiondf-coels 35696 Define the class of coelements on the class 𝐴, see also the alternate definition dfcoels 35711. Possible definitions are the special cases of dfcoss3 35698 and dfcoss4 35699. (Contributed by Peter Mazsa, 20-Nov-2019.)
𝐴 = ≀ ( E ↾ 𝐴)

Theoremdfcoss2 35697* Alternate definition of the class of cosets by 𝑅: 𝑥 and 𝑦 are cosets by 𝑅 iff there exists a set 𝑢 such that both 𝑥 and 𝑦 are are elements of the 𝑅-coset of 𝑢 (see also the comment of dfec2 8270). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅)}

Theoremdfcoss3 35698 Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 35695). (Contributed by Peter Mazsa, 27-Dec-2018.)
𝑅 = (𝑅𝑅)

Theoremdfcoss4 35699 Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 35695). (Contributed by Peter Mazsa, 12-Jul-2021.)
𝑅 = ran (𝑅𝑅)

Theoremcossex 35700 If 𝐴 is a set then the class of cosets by 𝐴 is a set. (Contributed by Peter Mazsa, 4-Jan-2019.)
(𝐴𝑉 → ≀ 𝐴 ∈ V)

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