P. 345 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34401-34500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ecres 34401*	Restricted coset of 𝐵. (Contributed by Peter Mazsa, 9-Dec-2018.)
⊢ [𝐵](𝑅 ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)}
Theorem	ecres2 34402	The restricted coset of 𝐵 when 𝐵 is an element of the restriction. (Contributed by Peter Mazsa, 16-Oct-2018.)
⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)
Theorem	eccnvepres 34403*	Restricted converse epsilon coset of 𝐵. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 21-Oct-2021.)
⊢ (𝐵 ∈ 𝑉 → [𝐵](◡ E ↾ 𝐴) = {𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴})
Theorem	eleccnvep 34404	Elementhood in the converse epsilon coset of 𝐴 is elementhood in 𝐴. (Contributed by Peter Mazsa, 27-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ [𝐴]◡ E ↔ 𝐵 ∈ 𝐴))
Theorem	eccnvep 34405	The converse epsilon coset of a set is the set. (Contributed by Peter Mazsa, 27-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)
Theorem	extep 34406	Property of epsilon relation, cf. extid 34439, extssr 34616 and the comment of df-ssr 34605. (Contributed by Peter Mazsa, 10-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ E = [𝐵]◡ E ↔ 𝐴 = 𝐵))
Theorem	eccnvepres2 34407	The restricted converse epsilon coset of an element of the restriction is the element itself. (Contributed by Peter Mazsa, 16-Jul-2019.)
⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = 𝐵)
Theorem	eccnvepres3 34408	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)
Theorem	eldmqsres 34409*	Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 21-Aug-2020.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)))
Theorem	eldmqsres2 34410*	Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 22-Aug-2020.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅))
Theorem	qsss1 34411	Subclass theorem for quotient sets. (Contributed by Peter Mazsa, 12-Sep-2020.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶))
Theorem	qseq1i 34412	Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 / 𝐶) = (𝐵 / 𝐶)
Theorem	qseq1d 34413	Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶))
Theorem	qseq2i 34414	Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 / 𝐴) = (𝐶 / 𝐵)
Theorem	qseq2d 34415	Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 / 𝐴) = (𝐶 / 𝐵))
Theorem	qseq12 34416	Equality theorem for quotient set. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))
Theorem	brinxprnres 34417	Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.)
⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))
Theorem	inxprnres 34418*	Restriction of a class as a class of ordered pairs. (Contributed by Peter Mazsa, 2-Jan-2019.)
⊢ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}
Theorem	dfres4 34419	Alternate definition of the restriction of a class. (Contributed by Peter Mazsa, 2-Jan-2019.)
⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))
Theorem	exan3 34420*	Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅 ∧ 𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
Theorem	exanres 34421*	Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 2-May-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢(𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
Theorem	exanres3 34422*	Equivalent expressions with restricted existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢 ∈ 𝐴 (𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
Theorem	exanres2 34423*	Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢(𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆)))
Theorem	cnvepres 34424*	Restricted converse epsilon relation as a class of ordered pairs. (Contributed by Peter Mazsa, 10-Feb-2018.)
⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
Theorem	ssrel3 34425*	Subclass relation in another form when the subclass is a relation. (Contributed by Peter Mazsa, 16-Feb-2019.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦)))
Theorem	eqrel2 34426*	Equality of relations. (Contributed by Peter Mazsa, 8-Mar-2019.)
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)))
Theorem	relinxp 34427	Intersection with a Cartesian product is a relation. (Contributed by Peter Mazsa, 4-Mar-2019.)
⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
Theorem	rncnv 34428	Range of converse is the domain. (Contributed by Peter Mazsa, 12-Feb-2018.)
⊢ ran ◡𝐴 = dom 𝐴
Theorem	dfdm6 34429*	Alternate definition of domain. (Contributed by Peter Mazsa, 2-Mar-2018.)
⊢ dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
Theorem	dfrn6 34430*	Alternate definition of range. (Contributed by Peter Mazsa, 1-Aug-2018.)
⊢ ran 𝑅 = {𝑥 ∣ [𝑥]◡𝑅 ≠ ∅}
Theorem	rncnvepres 34431	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
Theorem	dmecd 34432	Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 7963). (Contributed by Peter Mazsa, 9-Oct-2018.)
⊢ (𝜑 → dom 𝑅 = 𝐴)    &   ⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)    ⇒   ⊢ (𝜑 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
Theorem	dmec2d 34433	Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 7963). (Contributed by Peter Mazsa, 12-Oct-2018.)
⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)    ⇒   ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅))
Theorem	inxpssres 34434	Intersection with a Cartesian product is a subclass of restriction. (Contributed by Peter Mazsa, 19-Jul-2019.)
⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴)
Theorem	brid 34435	Property of the identity binary relation. (Contributed by Peter Mazsa, 18-Dec-2021.)
⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴)
Theorem	ideq2 34436	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.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	idresssidinxp 34437	Condition for the identity restriction to be a subclass of identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018.)
⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))
Theorem	idreseqidinxp 34438	Condition for the identity restriction to be equal to the identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018.)
⊢ (𝐴 ⊆ 𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴))
Theorem	extid 34439	Property of identity relation, cf. extep 34406, extssr 34616 and the comment of df-ssr 34605. (Contributed by Peter Mazsa, 5-Jul-2019.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))
Theorem	inxpss 34440*	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 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥𝑆𝑦))
Theorem	idinxpss 34441*	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 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦))
Theorem	inxpss3 34442*	Two ways to say that an intersection with a Cartesian product is a subclass (cf. inxpss 34440). (Contributed by Peter Mazsa, 8-Mar-2019.)
⊢ (∀𝑥∀𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 → 𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥𝑆𝑦))
Theorem	inxpss2 34443*	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)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥𝑆𝑦))
Theorem	inxpssidinxp 34444*	Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 34443. (Contributed by Peter Mazsa, 4-Jul-2019.)
⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 = 𝑦))
Theorem	idinxpssinxp 34445*	Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 34443. (Contributed by Peter Mazsa, 6-Mar-2019.)
⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦))
Theorem	idinxpres 34446	The intersection of the identity function with a square cross product. (Contributed by FL, 2-Aug-2009.)
⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
Theorem	idinxpssinxp2 34447*	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)) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
Theorem	idinxpssinxp3 34448	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)
Theorem	idinxpssinxp4 34449*	Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (cf. idinxpssinxp2 34447). (Contributed by Peter Mazsa, 8-Mar-2019.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
Theorem	relcnveq3 34450*	Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.)
⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
Theorem	relcnveq 34451	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.)
⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))
Theorem	relcnveq2 34452*	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)
⊢ (Rel 𝑅 → (◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))
Theorem	relcnveq4 34453*	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)
⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))
Theorem	qsresid 34454	Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.)
⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅)
Theorem	nel02 34455	The empty set has no elements. (Contributed by Peter Mazsa, 4-Jan-2018.)
⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴)
Theorem	n0elqs 34456	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)
Theorem	n0elqs2 34457	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 (𝑅 ↾ 𝐴) = 𝐴)
Theorem	ecex2 34458	Condition for a coset to be a set. (Contributed by Peter Mazsa, 4-May-2019.)
⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ V))
Theorem	uniqsALTV 34459	The union of a quotient set: a weaker version of uniqs 7980. (Contributed by Peter Mazsa, 20-Jun-2019.)
⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))
Theorem	rnresequniqs 34460	The range of a restriction is equal to the union of the quotient set. (Contributed by Peter Mazsa, 19-May-2018.)
⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ran (𝑅 ↾ 𝐴) = ∪ (𝐴 / 𝑅))
Theorem	n0el2 34461	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)
Theorem	cnvepresex 34462	Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.)
⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V)
Theorem	inex2ALTV 34463	Sufficient condition for an intersection relation to be a set, cf. inex1g 4949. (Contributed by Peter Mazsa, 19-Dec-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V)
Theorem	inex3 34464	Sufficient condition for the intersection relation to be a set. (Contributed by Peter Mazsa, 24-Nov-2019.)
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V)
Theorem	inxpex 34465	Sufficient condition for an intersection with a Cartesian product to be a set. (Contributed by Peter Mazsa, 10-May-2019.)
⊢ ((𝑅 ∈ 𝑊 ∨ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
Theorem	eqres 34466	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 = (𝑆 ↾ 𝐶)    ⇒   ⊢ (𝐵 ∈ 𝑉 → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵)))
Theorem	opideq 34467	Equality conditions for ordered pairs ⟨𝐴, 𝐴⟩ and ⟨𝐵, 𝐵⟩. (Contributed by Peter Mazsa, 22-Jul-2019.) (Revised by Thierry Arnoux, 16-Feb-2022.)
⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ 𝐴 = 𝐵))
Theorem	opelinxp 34468	Ordered pair element in an intersection with Cartesian product. (Contributed by Peter Mazsa, 21-Jul-2019.)
⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))
Theorem	iss2 34469	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)))
Theorem	eldmcnv 34470*	Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Theorem	dfrel5 34471	Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 6-Nov-2018.)
⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
Theorem	dfrel6 34472	Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.)
⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
Theorem	cnvresrn 34473	Converse restricted to range is converse. (Contributed by Peter Mazsa, 3-Sep-2021.)
⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅
Theorem	ecin0 34474*	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
Theorem	ecinn0 34475*	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
Theorem	ineleq 34476*	Equivalence of restricted universal quantifications. (Contributed by Peter Mazsa, 29-May-2018.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧∀𝑦 ∈ 𝐵 ((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦))
Theorem	inecmo 34477*	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 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (Rel 𝑅 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧))
Theorem	inecmo2 34478*	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.)
⊢ ((∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥 ∧ Rel 𝑅))
Theorem	ineccnvmo 34479*	Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.)
⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)
Theorem	alrmomorn 34480	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)
Theorem	alrmomodm 34481*	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))
Theorem	ineccnvmo2 34482*	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 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)
Theorem	inecmo3 34483*	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 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅))
Theorem	moantr 34484	Sufficient condition for transitivity of conjunctions inside existential quantifiers. (Contributed by Peter Mazsa, 2-Oct-2018.)
⊢ (∃*𝑥𝜓 → ((∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜓 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜒)))
Theorem	brabidga 34485	The law of concretion for a binary relation. Special case of brabga 5136. (Contributed by Peter Mazsa, 24-Nov-2018.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ (𝑥𝑅𝑦 ↔ 𝜑)
Theorem	inxp2 34486*	Intersection with a Cartesian product. (Contributed by Peter Mazsa, 18-Jul-2019.)
⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)}
Theorem	opabf 34487	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} = ∅
Theorem	ec0 34488	The empty-coset of a class is the empty set. (Contributed by Peter Mazsa, 19-May-2019.)
⊢ [𝐴]∅ = ∅
Theorem	0qs 34489	Quotient set with the empty set. (Contributed by Peter Mazsa, 14-Sep-2019.)
⊢ (∅ / 𝑅) = ∅
20.21.3  Range Cartesian product
Definition	df-xrn 34490	Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 34491 and brxrn 34493. This is Scott Fenton's df-txp 32315 with a different symbol, cf. https://github.com/metamath/set.mm/issues/2469. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
Theorem	xrnss3v 34491	A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 32339 with a different symbol, cf. https://github.com/metamath/set.mm/issues/2469. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
Theorem	xrnrel 34492	A range Cartesian product is a relation. This is Scott Fenton's txprel 32340 with a different symbol, cf. https://github.com/metamath/set.mm/issues/2469. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Rel (𝐴 ⋉ 𝐵)
Theorem	brxrn 34493	Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 34491, this characterizes elementhood in a range cross. (Contributed by Peter Mazsa, 27-Jun-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐶)))
Theorem	brxrn2 34494*	A characterization of the range Cartesian product. (Contributed by Peter Mazsa, 14-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
Theorem	dfxrn2 34495*	Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022.)
⊢ (𝑅 ⋉ 𝑆) = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
Theorem	xrneq1 34496	Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))
Theorem	xrneq1i 34497	Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)
Theorem	xrneq1d 34498	Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))
Theorem	xrneq2 34499	Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))
Theorem	xrneq2i 34500	Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵)