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Theorem List for Metamath Proof Explorer - 34201-34300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoreminxprnres 34201* Restriction of a class as a class of ordered pairs. (Contributed by Peter Mazsa, 2-Jan-2019.)
(𝑅 ∩ (𝐴 × ran (𝑅𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}

Theoremdfres4 34202 Alternate definition of the restriction of a class. (Contributed by Peter Mazsa, 2-Jan-2019.)
(𝑅𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))

Theoremexan3 34203* Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.)
((𝐴𝑉𝐵𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))

Theoremexanres 34204* Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 2-May-2021.)
((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))

Theoremexanres3 34205* Equivalent expressions with restricted existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.)
((𝐵𝑉𝐶𝑊) → (∃𝑢𝐴 (𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))

Theoremexanres2 34206* Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.)
((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃𝑢𝐴 (𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆)))

Theoremcnvepres 34207* Restricted converse epsilon relation as a class of ordered pairs. (Contributed by Peter Mazsa, 10-Feb-2018.)
( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}

Theoremssrel3 34208* Subclass relation in another form when the subclass is a relation. (Contributed by Peter Mazsa, 16-Feb-2019.)
(Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))

Theoremeqrel2 34209* Equality of relations. (Contributed by Peter Mazsa, 8-Mar-2019.)
((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))

Theoremrelinxp 34210 Intersection with a Cartesian product is a relation. (Contributed by Peter Mazsa, 4-Mar-2019.)
Rel (𝑅 ∩ (𝐴 × 𝐵))

Theoremrncnv 34211 Range of converse is the domain. (Contributed by Peter Mazsa, 12-Feb-2018.)
ran 𝐴 = dom 𝐴

Theoremdfdm6 34212* Alternate definition of domain. (Contributed by Peter Mazsa, 2-Mar-2018.)
dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}

Theoremdfrn6 34213* Alternate definition of range. (Contributed by Peter Mazsa, 1-Aug-2018.)
ran 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}

Theoremrncnvepres 34214 The range of the restricted converse epsilon is the union of the restriction. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 26-Sep-2021.)
ran ( E ↾ 𝐴) = 𝐴

Theoremdmecd 34215 Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 7833). (Contributed by Peter Mazsa, 9-Oct-2018.)
(𝜑 → dom 𝑅 = 𝐴)    &   (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)       (𝜑 → (𝐵𝐴𝐶𝐴))

Theoremdmec2d 34216 Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 7833). (Contributed by Peter Mazsa, 12-Oct-2018.)
(𝜑 → [𝐵]𝑅 = [𝐶]𝑅)       (𝜑 → (𝐵 ∈ dom 𝑅𝐶 ∈ dom 𝑅))

Theoreminxpssres 34217 Intersection with a Cartesian product is a subclass of restriction. (Contributed by Peter Mazsa, 19-Jul-2019.)
(𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅𝐴)

Theorembrid 34218 Property of the identity binary relation. (Contributed by Peter Mazsa, 18-Dec-2021.)
(𝐴 I 𝐵𝐵 I 𝐴)

Theoremideq2 34219 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 34220 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 34221 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 34222 Property of identity relation, cf. extep 34189, extssr 34399 and the comment of df-ssr 34388. (Contributed by Peter Mazsa, 5-Jul-2019.)
(𝐴𝑉 → ([𝐴] I = [𝐵] I ↔ 𝐴 = 𝐵))

Theoreminxpss 34223* 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 34224* 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 34225* Two ways to say that an intersection with a Cartesian product is a subclass (cf. inxpss 34223). (Contributed by Peter Mazsa, 8-Mar-2019.)
(∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))

Theoreminxpss2 34226* 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 34227* Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 34226. (Contributed by Peter Mazsa, 4-Jul-2019.)
((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))

Theoremidinxpssinxp 34228* Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 34226. (Contributed by Peter Mazsa, 6-Mar-2019.)
(( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))

Theoremidinxpres 34229 The intersection of the identity function with a square cross product. (Contributed by FL, 2-Aug-2009.)
( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)

Theoremidinxpssinxp2 34230* 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.)
(( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)

Theoremidinxpssinxp3 34231 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.)
(( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ( I ↾ 𝐴) ⊆ 𝑅)

Theoremidinxpssinxp4 34232* Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (cf. idinxpssinxp2 34230). (Contributed by Peter Mazsa, 8-Mar-2019.)
(∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)

Theoremrelcnveq3 34233* Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.)
(Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

Theoremrelcnveq 34234 Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.)
(Rel 𝑅 → (𝑅𝑅𝑅 = 𝑅))

Theoremrelcnveq2 34235* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)
(Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

Theoremrelcnveq4 34236* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)
(Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

Theoremqsresid 34237 Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.)
(𝐴 / (𝑅𝐴)) = (𝐴 / 𝑅)

Theoremnel02 34238 The empty set has no elements. (Contributed by Peter Mazsa, 4-Jan-2018.)
(𝐴 = ∅ → ¬ 𝐵𝐴)

Theoremn0elqs 34239 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 34240 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 34241 Condition for a coset to be a set. (Contributed by Peter Mazsa, 4-May-2019.)
((𝑅𝐴) ∈ 𝑉 → (𝐵𝐴 → [𝐵]𝑅 ∈ V))

TheoremuniqsALTV 34242 The union of a quotient set: a weaker version of uniqs 7850. (Contributed by Peter Mazsa, 20-Jun-2019.)
((𝑅𝐴) ∈ 𝑉 (𝐴 / 𝑅) = (𝑅𝐴))

Theoremrnresequniqs 34243 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 34244 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 34245 Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.)
(𝐴𝑉 → ( E ↾ 𝐴) ∈ V)

Theoreminex2ALTV 34246 Sufficient condition for an intersection relation to be a set, cf. inex1g 4834. (Contributed by Peter Mazsa, 19-Dec-2018.)
(𝐴𝑉 → (𝐵𝐴) ∈ V)

Theoreminex3 34247 Sufficient condition for the intersection relation to be a set. (Contributed by Peter Mazsa, 24-Nov-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Theoreminxpex 34248 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 34249 Converting a class constant definition by restriction (like ~? df-ers or ~? df-parts ) into a binary relation. (Contributed by Peter Mazsa, 1-Oct-2018.)
𝑅 = (𝑆𝐶)       (𝐵𝑉 → (𝐴𝑅𝐵 ↔ (𝐴𝐶𝐴𝑆𝐵)))

Theoremopideq 34250 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))

Theoremopelinxp 34251 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))

Theoremiss2 34252 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 34253* Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.)
(𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

Theoremdfrel5 34254 Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 6-Nov-2018.)
(Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)

Theoremdfrel6 34255 Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.)
(Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)

Theoremcnvresrn 34256 Converse restricted to range is converse. (Contributed by Peter Mazsa, 3-Sep-2021.)
(𝑅 ↾ ran 𝑅) = 𝑅

Theoremecin0 34257* 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 34258* 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 34259* Equivalence of restricted universal quantifications. (Contributed by Peter Mazsa, 29-May-2018.)
(∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))

Theoreminecmo 34260* 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 34261* 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 34262* 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 34263 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 34264* 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 34265* 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 34266* 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 34267 Sufficient condition for transitivity of conjunctions inside existential quantifiers. (Contributed by Peter Mazsa, 2-Oct-2018.)
(∃*𝑥𝜓 → ((∃𝑥(𝜑𝜓) ∧ ∃𝑥(𝜓𝜒)) → ∃𝑥(𝜑𝜒)))

Theorembrabidga 34268 The law of concretion for a binary relation. Special case of brabga 5018. (Contributed by Peter Mazsa, 24-Nov-2018.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       (𝑥𝑅𝑦𝜑)

Theoreminxp2 34269* Intersection with a Cartesian product. (Contributed by Peter Mazsa, 18-Jul-2019.)
(𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦)}

Theoremopabf 34270 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 34271 The empty-coset of a class is the empty set. (Contributed by Peter Mazsa, 19-May-2019.)
[𝐴]∅ = ∅

Theorem0qs 34272 Quotient set with the empty set. (Contributed by Peter Mazsa, 14-Sep-2019.)
(∅ / 𝑅) = ∅

20.21.3  Range Cartesian product

Definitiondf-xrn 34273 Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 34274 and brxrn 34276. This is Scott Fenton's df-txp 32086 with a different symbol, cf. https://github.com/metamath/set.mm/issues/2469. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))

Theoremxrnss3v 34274 A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 32110 with a different symbol, cf. https://github.com/metamath/set.mm/issues/2469. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) ⊆ (V × (V × V))

Theoremxrnrel 34275 A range Cartesian product is a relation. This is Scott Fenton's txprel 32111 with a different symbol, cf. https://github.com/metamath/set.mm/issues/2469. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel (𝐴𝐵)

Theorembrxrn 34276 Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 34274, this characterizes elementhood in a range cross. (Contributed by Peter Mazsa, 27-Jun-2021.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵𝐴𝑆𝐶)))

Theorembrxrn2 34277* A characterization of the range Cartesian product. (Contributed by Peter Mazsa, 14-Oct-2020.)
(𝐴𝑉 → (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))

Theoremdfxrn2 34278* Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022.)
(𝑅𝑆) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥𝑢𝑆𝑦)}

Theoremxrneq1 34279 Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremxrneq1i 34280 Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremxrneq1d 34281 Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremxrneq2 34282 Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremxrneq2i 34283 Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremxrneq2d 34284 Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremxrneq12 34285 Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremxrneq12i 34286 Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremxrneq12d 34287 Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremelecxrn 34288* 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 34289* 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 34290* 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 34291* 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 34292 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 34293 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 34294* 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 34295* 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 34296* Range of the range Cartesian product of classes. (Contributed by Peter Mazsa, 1-Jun-2020.)
ran (𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)}

Theoremrnxrnres 34297* 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 34298* 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 34299* 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 34300 Two ways to express restriction of range Cartesian product, cf. xrnres2 34301, xrnres3 34302. (Contributed by Peter Mazsa, 5-Jun-2021.)
((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ 𝑆)

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