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Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabeqinbi 36401* Intersection with class abstraction and equivalent wff's. (Contributed by Peter Mazsa, 21-Jul-2021.)
𝐴 = (𝐵𝐶)    &   𝐵 = {𝑥𝜑}    &   (𝑥𝐶 → (𝜑𝜓))       𝐴 = {𝑥𝐶𝜓}
 
Theoremrabeqel 36402* Class element of a restricted class abstraction. (Contributed by Peter Mazsa, 24-Jul-2021.)
𝐵 = {𝑥𝐴𝜑}    &   (𝑥 = 𝐶 → (𝜑𝜓))       (𝐶𝐵 ↔ (𝜓𝐶𝐴))
 
Theoremeqrelf 36403* The equality connective between relations. (Contributed by Peter Mazsa, 25-Jun-2019.)
𝑥𝐴    &   𝑥𝐵    &   𝑦𝐴    &   𝑦𝐵       ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
 
Theoremreleleccnv 36404 Elementhood in a converse 𝑅-coset when 𝑅 is a relation. (Contributed by Peter Mazsa, 9-Dec-2018.)
(Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐴𝑅𝐵))
 
Theoremreleccnveq 36405* Equality of converse 𝑅-coset and converse 𝑆-coset when 𝑅 and 𝑆 are relations. (Contributed by Peter Mazsa, 27-Jul-2019.)
((Rel 𝑅 ∧ Rel 𝑆) → ([𝐴]𝑅 = [𝐵]𝑆 ↔ ∀𝑥(𝑥𝑅𝐴𝑥𝑆𝐵)))
 
Theoremopelvvdif 36406 Negated elementhood of ordered pair. (Contributed by Peter Mazsa, 14-Jan-2019.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ ((V × V) ∖ 𝑅) ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
 
Theoremvvdifopab 36407* Ordered-pair class abstraction defined by a negation. (Contributed by Peter Mazsa, 25-Jun-2019.)
((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
 
Theorembrvdif 36408 Binary relation with universal complement is the negation of the relation. (Contributed by Peter Mazsa, 1-Jul-2018.)
(𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)
 
Theorembrvdif2 36409 Binary relation with universal complement. (Contributed by Peter Mazsa, 14-Jul-2018.)
(𝐴(V ∖ 𝑅)𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
 
Theorembrvvdif 36410 Binary relation with the complement under the universal class of ordered pairs. (Contributed by Peter Mazsa, 9-Nov-2018.)
((𝐴𝑉𝐵𝑊) → (𝐴((V × V) ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵))
 
Theorembrvbrvvdif 36411 Binary relation with the complement under the universal class of ordered pairs is the same as with universal complement. (Contributed by Peter Mazsa, 28-Nov-2018.)
((𝐴𝑉𝐵𝑊) → (𝐴((V × V) ∖ 𝑅)𝐵𝐴(V ∖ 𝑅)𝐵))
 
Theorembrcnvep 36412 The converse of the binary epsilon relation. (Contributed by Peter Mazsa, 30-Jan-2018.)
(𝐴𝑉 → (𝐴 E 𝐵𝐵𝐴))
 
TheoremelecALTV 36413 Elementhood in the 𝑅-coset of 𝐴. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 8550 with this original form of Suppes. Peter Mazsa). (Contributed by Mario Carneiro, 9-Jul-2014.)
((𝐴𝑉𝐵𝑊) → (𝐵 ∈ [𝐴]𝑅𝐴𝑅𝐵))
 
Theorembrcnvepres 36414 Restricted converse epsilon binary relation. (Contributed by Peter Mazsa, 10-Feb-2018.)
((𝐵𝑉𝐶𝑊) → (𝐵( E ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐶𝐵)))
 
Theorembrres2 36415 Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.) (Revised by Peter Mazsa, 16-Dec-2021.)
(𝐵(𝑅𝐴)𝐶𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶)
 
Theoremeldmres 36416* Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 9-Jan-2019.)
(𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦)))
 
Theoremeldm4 36417* Elementhood in a domain. (Contributed by Peter Mazsa, 24-Oct-2018.)
(𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅))
 
Theoremeldmres2 36418* Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 21-Aug-2020.)
(𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ (𝐵𝐴 ∧ ∃𝑦 𝑦 ∈ [𝐵]𝑅)))
 
Theoremeceq1i 36419 Equality theorem for 𝐶-coset of 𝐴 and 𝐶-coset of 𝐵, inference version. (Contributed by Peter Mazsa, 11-May-2021.)
𝐴 = 𝐵       [𝐴]𝐶 = [𝐵]𝐶
 
Theoremelecres 36420 Elementhood in the restricted coset of 𝐵. (Contributed by Peter Mazsa, 21-Sep-2018.)
(𝐶𝑉 → (𝐶 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝐶)))
 
Theoremecres 36421* Restricted coset of 𝐵. (Contributed by Peter Mazsa, 9-Dec-2018.)
[𝐵](𝑅𝐴) = {𝑥 ∣ (𝐵𝐴𝐵𝑅𝑥)}
 
Theoremecres2 36422 The restricted coset of 𝐵 when 𝐵 is an element of the restriction. (Contributed by Peter Mazsa, 16-Oct-2018.)
(𝐵𝐴 → [𝐵](𝑅𝐴) = [𝐵]𝑅)
 
Theoremeccnvepres 36423* Restricted converse epsilon coset of 𝐵. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 21-Oct-2021.)
(𝐵𝑉 → [𝐵]( E ↾ 𝐴) = {𝑥𝐵𝐵𝐴})
 
Theoremeleccnvep 36424 Elementhood in the converse epsilon coset of 𝐴 is elementhood in 𝐴. (Contributed by Peter Mazsa, 27-Jan-2019.)
(𝐴𝑉 → (𝐵 ∈ [𝐴] E ↔ 𝐵𝐴))
 
Theoremeccnvep 36425 The converse epsilon coset of a set is the set. (Contributed by Peter Mazsa, 27-Jan-2019.)
(𝐴𝑉 → [𝐴] E = 𝐴)
 
Theoremextep 36426 Property of epsilon relation, see also extid 36453, extssr 36634 and the comment of df-ssr 36623. (Contributed by Peter Mazsa, 10-Jul-2019.)
((𝐴𝑉𝐵𝑊) → ([𝐴] E = [𝐵] E ↔ 𝐴 = 𝐵))
 
Theoremeccnvepres2 36427 The restricted converse epsilon coset of an element of the restriction is the element itself. (Contributed by Peter Mazsa, 16-Jul-2019.)
(𝐵𝐴 → [𝐵]( E ↾ 𝐴) = 𝐵)
 
Theoremeccnvepres3 36428 Condition for a restricted converse epsilon coset of a set to be the set itself. (Contributed by Peter Mazsa, 11-May-2021.)
(𝐵 ∈ dom ( E ↾ 𝐴) → [𝐵]( E ↾ 𝐴) = 𝐵)
 
Theoremeldmqsres 36429* Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 21-Aug-2020.)
(𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
 
Theoremeldmqsres2 36430* Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 22-Aug-2020.)
(𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅))
 
Theoremqsss1 36431 Subclass theorem for quotient sets. (Contributed by Peter Mazsa, 12-Sep-2020.)
(𝐴𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶))
 
Theoremqseq1i 36432 Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.)
𝐴 = 𝐵       (𝐴 / 𝐶) = (𝐵 / 𝐶)
 
Theoremqseq1d 36433 Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶))
 
Theorembrinxprnres 36434 Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.)
(𝐶𝑉 → (𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
 
Theoreminxprnres 36435* 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 36436 Alternate definition of the restriction of a class. (Contributed by Peter Mazsa, 2-Jan-2019.)
(𝑅𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
 
Theoremexan3 36437* Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.)
((𝐴𝑉𝐵𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
 
Theoremexanres 36438* Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 2-May-2021.)
((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))
 
Theoremexanres3 36439* Equivalent expressions with restricted existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.)
((𝐵𝑉𝐶𝑊) → (∃𝑢𝐴 (𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))
 
Theoremexanres2 36440* 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 36441* Restricted converse epsilon relation as a class of ordered pairs. (Contributed by Peter Mazsa, 10-Feb-2018.)
( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
 
Theoremeqrel2 36442* Equality of relations. (Contributed by Peter Mazsa, 8-Mar-2019.)
((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))
 
Theoremrncnv 36443 Range of converse is the domain. (Contributed by Peter Mazsa, 12-Feb-2018.)
ran 𝐴 = dom 𝐴
 
Theoremdfdm6 36444* Alternate definition of domain. (Contributed by Peter Mazsa, 2-Mar-2018.)
dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
 
Theoremdfrn6 36445* Alternate definition of range. (Contributed by Peter Mazsa, 1-Aug-2018.)
ran 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
 
Theoremrncnvepres 36446 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 36447 Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 8555). (Contributed by Peter Mazsa, 9-Oct-2018.)
(𝜑 → dom 𝑅 = 𝐴)    &   (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)       (𝜑 → (𝐵𝐴𝐶𝐴))
 
Theoremdmec2d 36448 Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 8555). (Contributed by Peter Mazsa, 12-Oct-2018.)
(𝜑 → [𝐵]𝑅 = [𝐶]𝑅)       (𝜑 → (𝐵 ∈ dom 𝑅𝐶 ∈ dom 𝑅))
 
Theorembrid 36449 Property of the identity binary relation. (Contributed by Peter Mazsa, 18-Dec-2021.)
(𝐴 I 𝐵𝐵 I 𝐴)
 
Theoremideq2 36450 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 36451 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 36452 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 36453 Property of identity relation, see also extep 36426, extssr 36634 and the comment of df-ssr 36623. (Contributed by Peter Mazsa, 5-Jul-2019.)
(𝐴𝑉 → ([𝐴] I = [𝐵] I ↔ 𝐴 = 𝐵))
 
Theoreminxpss 36454* 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 36455* 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 36456* Two ways to say that an intersection with a Cartesian product is a subclass (see also inxpss 36454). (Contributed by Peter Mazsa, 8-Mar-2019.)
(∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
 
Theoreminxpss2 36457* 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 36458* Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 36457. (Contributed by Peter Mazsa, 4-Jul-2019.)
((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))
 
Theoremidinxpssinxp 36459* Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 36457. (Contributed by Peter Mazsa, 6-Mar-2019.)
(( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
 
Theoremidinxpssinxp2 36460* 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 36461 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 36462* Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (see also idinxpssinxp2 36460). (Contributed by Peter Mazsa, 8-Mar-2019.)
(∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
 
Theoremrelcnveq3 36463* Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.)
(Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
 
Theoremrelcnveq 36464 Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.)
(Rel 𝑅 → (𝑅𝑅𝑅 = 𝑅))
 
Theoremrelcnveq2 36465* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)
(Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
 
Theoremrelcnveq4 36466* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)
(Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
 
Theoremqsresid 36467 Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.)
(𝐴 / (𝑅𝐴)) = (𝐴 / 𝑅)
 
Theoremn0elqs 36468 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 36469 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 36470 Condition for a coset to be a set. (Contributed by Peter Mazsa, 4-May-2019.)
((𝑅𝐴) ∈ 𝑉 → (𝐵𝐴 → [𝐵]𝑅 ∈ V))
 
TheoremuniqsALTV 36471 The union of a quotient set, like uniqs 8575 but with a weaker antecedent: only the restricion of 𝑅 by 𝐴 needs to be a set, not 𝑅 itself, see e.g. cnvepima 36479. (Contributed by Peter Mazsa, 20-Jun-2019.)
((𝑅𝐴) ∈ 𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
 
TheoremimaexALTV 36472 Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7771) with weakened antecedent: only the restricion of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 36478. (Contributed by Peter Mazsa, 22-Feb-2023.) (Proof modification is discouraged.)
((𝐴𝑉 ∨ ((𝐴𝐵) ∈ 𝑊𝐵𝑋)) → (𝐴𝐵) ∈ V)
 
TheoremecexALTV 36473 Existence of a coset, like ecexg 8511 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 36477. (Contributed by Peter Mazsa, 22-Feb-2023.)
((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V)
 
Theoremrnresequniqs 36474 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 36475 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 36476 Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.)
(𝐴𝑉 → ( E ↾ 𝐴) ∈ V)
 
Theoremeccnvepex 36477 The converse epsilon coset exists. (Contributed by Peter Mazsa, 22-Mar-2023.)
[𝐴] E ∈ V
 
Theoremcnvepimaex 36478 The image of converse epsilon exists, proof via imaexALTV 36472 (see also cnvepima 36479 and uniexg 7602 for alternate way). (Contributed by Peter Mazsa, 22-Mar-2023.)
(𝐴𝑉 → ( E “ 𝐴) ∈ V)
 
Theoremcnvepima 36479 The image of converse epsilon. (Contributed by Peter Mazsa, 22-Mar-2023.)
(𝐴𝑉 → ( E “ 𝐴) = 𝐴)
 
Theoreminex3 36480 Sufficient condition for the intersection relation to be a set. (Contributed by Peter Mazsa, 24-Nov-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theoreminxpex 36481 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 36482 Converting a class constant definition by restriction (like df-ers 36782 or ~? df-parts ) into a binary relation. (Contributed by Peter Mazsa, 1-Oct-2018.)
𝑅 = (𝑆𝐶)       (𝐵𝑉 → (𝐴𝑅𝐵 ↔ (𝐴𝐶𝐴𝑆𝐵)))
 
Theorembrrabga 36483* 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 36484* 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 36485 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 36486 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 36487* Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.)
(𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
 
Theoremdfrel5 36488 Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 6-Nov-2018.)
(Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
 
Theoremdfrel6 36489 Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.)
(Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
 
Theoremcnvresrn 36490 Converse restricted to range is converse. (Contributed by Peter Mazsa, 3-Sep-2021.)
(𝑅 ↾ ran 𝑅) = 𝑅
 
Theoremecin0 36491* 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 36492* 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 36493* Equivalence of restricted universal quantifications. (Contributed by Peter Mazsa, 29-May-2018.)
(∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
 
Theoreminecmo 36494* 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 36495* 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 36496* 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 36497 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 36498* 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 36499* 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 36500* 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))
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