P. 356 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35501-35600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	el3v23 35501	New way (elv 3502, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ ((𝜑 ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	an2anr 35502	Double commutation in conjunction. (Contributed by Peter Mazsa, 27-Jun-2019.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜑) ∧ (𝜃 ∧ 𝜒)))
Theorem	anan 35503	Multiple commutations in conjunction. (Contributed by Peter Mazsa, 7-Mar-2020.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜑 ∧ 𝜃) ∧ 𝜏)) ↔ ((𝜓 ∧ 𝜃) ∧ (𝜑 ∧ (𝜒 ∧ 𝜏))))
Theorem	triantru3 35504	A wff is equivalent to its conjunctions with truths. (Contributed by Peter Mazsa, 30-Nov-2018.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ (𝜒 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	eqeltr 35505	Substitution of equal classes into elementhood relation. (Contributed by Peter Mazsa, 22-Jul-2017.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)
Theorem	eqelb 35506	Substitution of equal classes into elementhood relation. (Contributed by Peter Mazsa, 17-Jul-2019.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶))
Theorem	eqeqan2d 35507	Implication of introducing a new equality. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	ineqcom 35508	Two ways of saying that two classes are disjoint (when 𝐶 = ∅: ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)). (Contributed by Peter Mazsa, 22-Mar-2017.)
⊢ ((𝐴 ∩ 𝐵) = 𝐶 ↔ (𝐵 ∩ 𝐴) = 𝐶)
Theorem	ineqcomi 35509	Disjointness inference (when 𝐶 = ∅), inference form of ineqcom 35508. (Contributed by Peter Mazsa, 26-Mar-2017.)
⊢ (𝐴 ∩ 𝐵) = 𝐶    ⇒   ⊢ (𝐵 ∩ 𝐴) = 𝐶
Theorem	inres2 35510	Two ways of expressing the restriction of an intersection. (Contributed by Peter Mazsa, 5-Jun-2021.)
⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑅 ∩ 𝑆) ↾ 𝐴)
Theorem	coideq 35511	Equality theorem for composition of two classes. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐵))
Theorem	nexmo1 35512	If there is no case where wff is true, it is true for at most one case. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
Theorem	3albii 35513	Inference adding three universal quantifiers to both sides of an equivalence. (Contributed by Peter Mazsa, 10-Aug-2018.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥∀𝑦∀𝑧𝜑 ↔ ∀𝑥∀𝑦∀𝑧𝜓)
Theorem	3ralbii 35514	Inference adding three restricted universal quantifiers to both sides of an equivalence. (Contributed by Peter Mazsa, 25-Jul-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)
Theorem	ssrabi 35515	Inference of restricted abstraction subclass from implication. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabbieq 35516	Equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 8-Jul-2019.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabimbieq 35517	Restricted equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}    &   ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	abeqin 35518*	Intersection with class abstraction. (Contributed by Peter Mazsa, 21-Jul-2021.)
⊢ 𝐴 = (𝐵 ∩ 𝐶)    &   ⊢ 𝐵 = {𝑥 ∣ 𝜑}    ⇒   ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑}
Theorem	abeqinbi 35519*	Intersection with class abstraction and equivalent wff's. (Contributed by Peter Mazsa, 21-Jul-2021.)
⊢ 𝐴 = (𝐵 ∩ 𝐶)    &   ⊢ 𝐵 = {𝑥 ∣ 𝜑}    &   ⊢ (𝑥 ∈ 𝐶 → (𝜑 ↔ 𝜓))    ⇒   ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜓}
Theorem	rabeqel 35520*	Class element of a restricted class abstraction. (Contributed by Peter Mazsa, 24-Jul-2021.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐶 ∈ 𝐵 ↔ (𝜓 ∧ 𝐶 ∈ 𝐴))
Theorem	eqrelf 35521*	The equality connective between relations. (Contributed by Peter Mazsa, 25-Jun-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Theorem	releleccnv 35522	Elementhood in a converse 𝑅-coset when 𝑅 is a relation. (Contributed by Peter Mazsa, 9-Dec-2018.)
⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]◡𝑅 ↔ 𝐴𝑅𝐵))
Theorem	releccnveq 35523*	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(𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵)))
Theorem	opelvvdif 35524	Negated elementhood of ordered pair. (Contributed by Peter Mazsa, 14-Jan-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ ((V × V) ∖ 𝑅) ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
Theorem	vvdifopab 35525*	Ordered-pair class abstraction defined by a negation. (Contributed by Peter Mazsa, 25-Jun-2019.)
⊢ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
Theorem	brvdif 35526	Binary relation with universal complement is the negation of the relation. (Contributed by Peter Mazsa, 1-Jul-2018.)
⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)
Theorem	brvdif2 35527	Binary relation with universal complement. (Contributed by Peter Mazsa, 14-Jul-2018.)
⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
Theorem	brvvdif 35528	Binary relation with the complement under the universal class of ordered pairs. (Contributed by Peter Mazsa, 9-Nov-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴((V × V) ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵))
Theorem	brvbrvvdif 35529	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴((V × V) ∖ 𝑅)𝐵 ↔ 𝐴(V ∖ 𝑅)𝐵))
Theorem	brcnvep 35530	The converse of the binary epsilon relation. (Contributed by Peter Mazsa, 30-Jan-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝐵 ↔ 𝐵 ∈ 𝐴))
Theorem	elecALTV 35531	Elementhood in the 𝑅-coset of 𝐴. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 8335 with this original form of Suppes. Peter Mazsa) (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐵))
Theorem	brcnvepres 35532	Restricted converse epsilon binary relation. (Contributed by Peter Mazsa, 10-Feb-2018.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(◡ E ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)))
Theorem	brres2 35533	Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.) (Revised by Peter Mazsa, 16-Dec-2021.)
⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ 𝐵(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝐶)
Theorem	eldmres 35534*	Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 9-Jan-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝐵𝑅𝑦)))
Theorem	eldm4 35535*	Elementhood in a domain. (Contributed by Peter Mazsa, 24-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅))
Theorem	eldmres2 35536*	Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 21-Aug-2020.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ [𝐵]𝑅)))
Theorem	eceq1i 35537	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
Theorem	elecres 35538	Elementhood in the restricted coset of 𝐵. (Contributed by Peter Mazsa, 21-Sep-2018.)
⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))
Theorem	ecres 35539*	Restricted coset of 𝐵. (Contributed by Peter Mazsa, 9-Dec-2018.)
⊢ [𝐵](𝑅 ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)}
Theorem	ecres2 35540	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 35541*	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 35542	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 35543	The converse epsilon coset of a set is the set. (Contributed by Peter Mazsa, 27-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)
Theorem	extep 35544	Property of epsilon relation, see also extid 35572, extssr 35753 and the comment of df-ssr 35742. (Contributed by Peter Mazsa, 10-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ E = [𝐵]◡ E ↔ 𝐴 = 𝐵))
Theorem	eccnvepres2 35545	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 35546	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 35547*	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 35548*	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 35549	Subclass theorem for quotient sets. (Contributed by Peter Mazsa, 12-Sep-2020.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶))
Theorem	qseq1i 35550	Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 / 𝐶) = (𝐵 / 𝐶)
Theorem	qseq1d 35551	Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶))
Theorem	brinxprnres 35552	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 35553*	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 35554	Alternate definition of the restriction of a class. (Contributed by Peter Mazsa, 2-Jan-2019.)
⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))
Theorem	exan3 35555*	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 35556*	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 35557*	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 35558*	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 35559*	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 35560*	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 35561*	Equality of relations. (Contributed by Peter Mazsa, 8-Mar-2019.)
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)))
Theorem	rncnv 35562	Range of converse is the domain. (Contributed by Peter Mazsa, 12-Feb-2018.)
⊢ ran ◡𝐴 = dom 𝐴
Theorem	dfdm6 35563*	Alternate definition of domain. (Contributed by Peter Mazsa, 2-Mar-2018.)
⊢ dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
Theorem	dfrn6 35564*	Alternate definition of range. (Contributed by Peter Mazsa, 1-Aug-2018.)
⊢ ran 𝑅 = {𝑥 ∣ [𝑥]◡𝑅 ≠ ∅}
Theorem	rncnvepres 35565	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 35566	Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 8340). (Contributed by Peter Mazsa, 9-Oct-2018.)
⊢ (𝜑 → dom 𝑅 = 𝐴)    &   ⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)    ⇒   ⊢ (𝜑 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
Theorem	dmec2d 35567	Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 8340). (Contributed by Peter Mazsa, 12-Oct-2018.)
⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)    ⇒   ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅))
Theorem	brid 35568	Property of the identity binary relation. (Contributed by Peter Mazsa, 18-Dec-2021.)
⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴)
Theorem	ideq2 35569	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 35570	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 35571	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 35572	Property of identity relation, see also extep 35544, extssr 35753 and the comment of df-ssr 35742. (Contributed by Peter Mazsa, 5-Jul-2019.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))
Theorem	inxpss 35573*	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 35574*	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 35575*	Two ways to say that an intersection with a Cartesian product is a subclass (see also inxpss 35573). (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 35576*	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 35577*	Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 35576. (Contributed by Peter Mazsa, 4-Jul-2019.)
⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 = 𝑦))
Theorem	idinxpssinxp 35578*	Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 35576. (Contributed by Peter Mazsa, 6-Mar-2019.)
⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦))
Theorem	idinxpssinxp2 35579*	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)) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
Theorem	idinxpssinxp3 35580	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)
Theorem	idinxpssinxp4 35581*	Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (see also idinxpssinxp2 35579). (Contributed by Peter Mazsa, 8-Mar-2019.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
Theorem	relcnveq3 35582*	Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.)
⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
Theorem	relcnveq 35583	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.)
⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))
Theorem	relcnveq2 35584*	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 35585*	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 35586	Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.)
⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅)
Theorem	n0elqs 35587	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 35588	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 35589	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 35590	The union of a quotient set, like uniqs 8360 but with a weaker antecedent: only the restricion of 𝑅 by 𝐴 needs to be a set, not 𝑅 itself, see e.g. cnvepima 35598. (Contributed by Peter Mazsa, 20-Jun-2019.)
⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))
Theorem	imaexALTV 35591	Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7623) with weakened antecedent: only the restricion of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 35597. (Contributed by Peter Mazsa, 22-Feb-2023.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∨ ((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋)) → (𝐴 “ 𝐵) ∈ V)
Theorem	ecexALTV 35592	Existence of a coset, like ecexg 8296 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 35596. (Contributed by Peter Mazsa, 22-Feb-2023.)
⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V)
Theorem	rnresequniqs 35593	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 35594	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 35595	Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.)
⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V)
Theorem	eccnvepex 35596	The converse epsilon coset exists. (Contributed by Peter Mazsa, 22-Mar-2023.)
⊢ [𝐴]◡ E ∈ V
Theorem	cnvepimaex 35597	The image of converse epsilon exists, proof via imaexALTV 35591 (see also cnvepima 35598 and uniexg 7469 for alternate way). (Contributed by Peter Mazsa, 22-Mar-2023.)
⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) ∈ V)
Theorem	cnvepima 35598	The image of converse epsilon. (Contributed by Peter Mazsa, 22-Mar-2023.)
⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) = ∪ 𝐴)
Theorem	inex3 35599	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 35600	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)