P. 383 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38201-38300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	suceqsneq 38201	One-to-one relationship between the successor operation and the singleton. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = suc 𝐵 ↔ {𝐴} = {𝐵}))
Theorem	sucdifsn2 38202	Absorption of union with a singleton by difference. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
Theorem	sucdifsn 38203	The difference between the successor and the singleton of a class is the class. (Contributed by Peter Mazsa, 20-Sep-2024.)
⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴
Theorem	disjresin 38204	The restriction to a disjoint is the empty class. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = ∅)
Theorem	disjresdisj 38205	The intersection of restrictions to disjoint is the empty class. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∩ (𝑅 ↾ 𝐵)) = ∅)
Theorem	disjresdif 38206	The difference between restrictions to disjoint is the first restriction. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
Theorem	disjresundif 38207	Lemma for ressucdifsn2 38208. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
Theorem	ressucdifsn2 38208	The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn 38209. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)
Theorem	ressucdifsn 38209	The difference between restrictions to the successor and the singleton of a class is the restriction to the class. (Contributed by Peter Mazsa, 20-Sep-2024.)
⊢ ((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)
Theorem	inres2 38210	Two ways of expressing the restriction of an intersection. (Contributed by Peter Mazsa, 5-Jun-2021.)
⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑅 ∩ 𝑆) ↾ 𝐴)
Theorem	coideq 38211	Equality theorem for composition of two classes. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐵))
Theorem	nexmo1 38212	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	r2alan 38213*	Double restricted universal quantification, special case. (Contributed by Peter Mazsa, 17-Jun-2020.)
⊢ (∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓))
Theorem	ssrabi 38214	Inference of restricted abstraction subclass from implication. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabimbieq 38215	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 38216*	Intersection with class abstraction. (Contributed by Peter Mazsa, 21-Jul-2021.)
⊢ 𝐴 = (𝐵 ∩ 𝐶)    &   ⊢ 𝐵 = {𝑥 ∣ 𝜑}    ⇒   ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑}
Theorem	abeqinbi 38217*	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 38218*	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 38219*	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	br1cnvinxp 38220	Binary relation on the converse of an intersection with a Cartesian product. (Contributed by Peter Mazsa, 27-Jul-2019.)
⊢ (𝐶◡(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷𝑅𝐶))
Theorem	releleccnv 38221	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 38222*	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 38223	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 38224*	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 38225	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 38226	Binary relation with universal complement. (Contributed by Peter Mazsa, 14-Jul-2018.)
⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
Theorem	brvvdif 38227	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 38228	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 38229	The converse of the binary epsilon relation. (Contributed by Peter Mazsa, 30-Jan-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝐵 ↔ 𝐵 ∈ 𝐴))
Theorem	elecALTV 38230	Elementhood in the 𝑅-coset of 𝐴. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 8761 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 38231	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 38232	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	br1cnvres 38233	Binary relation on the converse of a restriction. (Contributed by Peter Mazsa, 27-Jul-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐵◡(𝑅 ↾ 𝐴)𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶𝑅𝐵)))
Theorem	eldmres 38234*	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	elrnres 38235*	Element of the range of a restriction. (Contributed by Peter Mazsa, 26-Dec-2018.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵))
Theorem	eldmressnALTV 38236	Element of the domain of a restriction to a singleton. (Contributed by Peter Mazsa, 12-Jun-2024.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 = 𝐴 ∧ 𝐴 ∈ dom 𝑅)))
Theorem	elrnressn 38237	Element of the range of a restriction to a singleton. (Contributed by Peter Mazsa, 12-Jun-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝐵))
Theorem	eldm4 38238*	Elementhood in a domain. (Contributed by Peter Mazsa, 24-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅))
Theorem	eldmres2 38239*	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 38240	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 38241	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 38242*	Restricted coset of 𝐵. (Contributed by Peter Mazsa, 9-Dec-2018.)
⊢ [𝐵](𝑅 ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)}
Theorem	ecres2 38243	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 38244*	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 38245	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 38246	The converse epsilon coset of a set is the set. (Contributed by Peter Mazsa, 27-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)
Theorem	extep 38247	Property of epsilon relation, see also extid 38274, extssr 38473 and the comment of df-ssr 38462. (Contributed by Peter Mazsa, 10-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ E = [𝐵]◡ E ↔ 𝐴 = 𝐵))
Theorem	disjeccnvep 38248	Property of the epsilon relation. (Contributed by Peter Mazsa, 27-Apr-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]◡ E ∩ [𝐵]◡ E ) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
Theorem	eccnvepres2 38249	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 38250	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 38251*	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 38252*	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 38253	Subclass theorem for quotient sets. (Contributed by Peter Mazsa, 12-Sep-2020.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶))
Theorem	qseq1i 38254	Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 / 𝐶) = (𝐵 / 𝐶)
Theorem	brinxprnres 38255	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 38256*	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 38257	Alternate definition of the restriction of a class. (Contributed by Peter Mazsa, 2-Jan-2019.)
⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))
Theorem	exan3 38258*	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 38259*	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 38260*	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 38261*	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 38262*	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	eqrel2 38263*	Equality of relations. (Contributed by Peter Mazsa, 8-Mar-2019.)
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)))
Theorem	rncnv 38264	Range of converse is the domain. (Contributed by Peter Mazsa, 12-Feb-2018.)
⊢ ran ◡𝐴 = dom 𝐴
Theorem	dfdm6 38265*	Alternate definition of domain. (Contributed by Peter Mazsa, 2-Mar-2018.)
⊢ dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
Theorem	dfrn6 38266*	Alternate definition of range. (Contributed by Peter Mazsa, 1-Aug-2018.)
⊢ ran 𝑅 = {𝑥 ∣ [𝑥]◡𝑅 ≠ ∅}
Theorem	rncnvepres 38267	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 38268	Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 8767). (Contributed by Peter Mazsa, 9-Oct-2018.)
⊢ (𝜑 → dom 𝑅 = 𝐴)    &   ⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)    ⇒   ⊢ (𝜑 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
Theorem	dmec2d 38269	Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 8767). (Contributed by Peter Mazsa, 12-Oct-2018.)
⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)    ⇒   ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅))
Theorem	brid 38270	Property of the identity binary relation. (Contributed by Peter Mazsa, 18-Dec-2021.)
⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴)
Theorem	ideq2 38271	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 38272	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 38273	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 38274	Property of identity relation, see also extep 38247, extssr 38473 and the comment of df-ssr 38462. (Contributed by Peter Mazsa, 5-Jul-2019.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))
Theorem	inxpss 38275*	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 38276*	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	ref5 38277*	Two ways to say that an intersection of the identity relation with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 12-Dec-2023.)
⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝑅𝑥)
Theorem	inxpss3 38278*	Two ways to say that an intersection with a Cartesian product is a subclass (see also inxpss 38275). (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 38279*	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 38280*	Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 38279. (Contributed by Peter Mazsa, 4-Jul-2019.)
⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 = 𝑦))
Theorem	idinxpssinxp 38281*	Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 38279. (Contributed by Peter Mazsa, 6-Mar-2019.)
⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦))
Theorem	idinxpssinxp2 38282*	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 38283	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 38284*	Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (see also idinxpssinxp2 38282). (Contributed by Peter Mazsa, 8-Mar-2019.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
Theorem	relcnveq3 38285*	Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.)
⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
Theorem	relcnveq 38286	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.)
⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))
Theorem	relcnveq2 38287*	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 38288*	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 38289	Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.)
⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅)
Theorem	n0elqs 38290	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 38291	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 38292	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 38293	The union of a quotient set, like uniqs 8789 but with a weaker antecedent: only the restriction of 𝑅 by 𝐴 needs to be a set, not 𝑅 itself, see e.g. cnvepima 38301. (Contributed by Peter Mazsa, 20-Jun-2019.)
⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))
Theorem	imaexALTV 38294	Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7907) with weakened antecedent: only the restriction of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 38300. (Contributed by Peter Mazsa, 22-Feb-2023.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∨ ((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋)) → (𝐴 “ 𝐵) ∈ V)
Theorem	ecexALTV 38295	Existence of a coset, like ecexg 8721 but with a weaker antecedent: only the restriction of 𝑅 by the singleton of 𝐴 needs to be a set, not 𝑅 itself, see e.g. eccnvepex 38299. (Contributed by Peter Mazsa, 22-Feb-2023.)
⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V)
Theorem	rnresequniqs 38296	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 38297	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 38298	Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.)
⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V)
Theorem	eccnvepex 38299	The converse epsilon coset exists. (Contributed by Peter Mazsa, 22-Mar-2023.)
⊢ [𝐴]◡ E ∈ V
Theorem	cnvepimaex 38300	The image of converse epsilon exists, proof via imaexALTV 38294 (see also cnvepima 38301 and uniexg 7732 for alternate way). (Contributed by Peter Mazsa, 22-Mar-2023.)
⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) ∈ V)