P. 388 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38701-38800   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	ccnvrefs 38701	Extend the definition of a class to include the converse reflexivity class.
class CnvRefs
Syntax	ccnvrefrels 38702	Extend the definition of a class to include the converse reflexive relations class.
class CnvRefRels
Syntax	wcnvrefrel 38703	Extend the definition of a wff to include the converse reflexive relation predicate. (Read: 𝑅 is a converse reflexive relation.)
wff CnvRefRel 𝑅
Syntax	csyms 38704	Extend the definition of a class to include the symmetry class.
class Syms
Syntax	csymrels 38705	Extend the definition of a class to include the symmetry relations class.
class SymRels
Syntax	wsymrel 38706	Extend the definition of a wff to include the symmetry relation predicate. (Read: 𝑅 is a symmetric relation.)
wff SymRel 𝑅
Syntax	ctrs 38707	Extend the definition of a class to include the transitivity class (but cf. the transitive class defined in df-tr 5213).
class Trs
Syntax	ctrrels 38708	Extend the definition of a class to include the transitive relations class.
class TrRels
Syntax	wtrrel 38709	Extend the definition of a wff to include the transitive relation predicate. (Read: 𝑅 is a transitive relation.)
wff TrRel 𝑅
Syntax	ceqvrels 38710	Extend the definition of a class to include the equivalence relations class.
class EqvRels
Syntax	weqvrel 38711	Extend the definition of a wff to include the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.)
wff EqvRel 𝑅
Syntax	ccoeleqvrels 38712	Extend the definition of a class to include the coelement equivalence relations class.
class CoElEqvRels
Syntax	wcoeleqvrel 38713	Extend the definition of a wff to include the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.)
wff CoElEqvRel 𝐴
Syntax	credunds 38714	Extend the definition of a class to include the redundancy class.
class Redunds
Syntax	wredund 38715	Extend the definition of a wff to include the redundancy predicate. (Read: 𝐴 is redundant with respect to 𝐵 in 𝐶.)
wff 𝐴 Redund ⟨𝐵, 𝐶⟩
Syntax	wredundp 38716	Extend wff definition to include the redundancy operator for propositions.
wff redund (𝜑, 𝜓, 𝜒)
Syntax	cdmqss 38717	Extend the definition of a class to include the domain quotients class.
class DomainQss
Syntax	wdmqs 38718	Extend the definition of a wff to include the domain quotient predicate. (Read: the domain quotient of 𝑅 is 𝐴.)
wff 𝑅 DomainQs 𝐴
Syntax	cers 38719	Extend the definition of a class to include the equivalence relations on their domain quotients class.
class Ers
Syntax	werALTV 38720	Extend the definition of a wff to include the equivalence relation on its domain quotient predicate. (Read: 𝑅 is an equivalence relation on its domain quotient 𝐴.)
wff 𝑅 ErALTV 𝐴
Syntax	cpeters 38721	Extend the definition of a class to include the blocklift-stable equivalence relations class.
class PetErs
Syntax	cpet2ers 38722	Extend the definition of a class to include the grade- and blocklift-stable equivalence relations class.
class Pet2Ers
Syntax	ccomembers 38723	Extend the definition of a class to include the comember equivalence relations class.
class CoMembErs
Syntax	wcomember 38724	Extend the definition of a wff to include the comember equivalence relation predicate. (Read: the comember equivalence relation on 𝐴, or, the restricted coelement equivalence relation on its domain quotient 𝐴.)
wff CoMembEr 𝐴
Syntax	cfunss 38725	Extend the definition of a class to include the function set class.
class Funss
Syntax	cfunsALTV 38726	Extend the definition of a class to include the functions class, i.e., the function relations class.
class FunsALTV
Syntax	wfunALTV 38727	Extend the definition of a wff to include the function predicate, i.e., the function relation predicate. (Read: 𝐹 is a function.)
wff FunALTV 𝐹
Syntax	cdisjss 38728	Extend the definition of a class to include the disjoint set class.
class Disjss
Syntax	cdisjs 38729	Extend the definition of a class to include the disjoints class, i.e., the disjoint relations class.
class Disjs
Syntax	wdisjALTV 38730	Extend the definition of a wff to include the disjoint predicate, i.e., the disjoint relation predicate. (Read: 𝑅 is a disjoint.)
wff Disj 𝑅
Syntax	celdisjs 38731	Extend the definition of a class to include the disjoint elements class, i.e., the disjoint element relations class.
class ElDisjs
Syntax	weldisj 38732	Extend the definition of a wff to include the disjoint element predicate, i.e., the disjoint element relation predicate. (Read: the elements of 𝐴 are disjoint.)
wff ElDisj 𝐴
Syntax	wantisymrel 38733	Extend the definition of a wff to include the antisymmetry relation predicate. (Read: 𝑅 is an antisymmetric relation.)
wff AntisymRel 𝑅
Syntax	cparts 38734	Extend the definition of a class to include the partitions class, i.e., the partition relations class.
class Parts
Syntax	wpart 38735	Extend the definition of a wff to include the partition predicate, i.e., the partition relation predicate. (Read: 𝐴 is a partition by 𝑅.)
wff 𝑅 Part 𝐴
Syntax	cmembparts 38736	Extend the definition of a class to include the member partitions class, i.e., the member partition relations class.
class MembParts
Syntax	wmembpart 38737	Extend the definition of a wff to include the member partition predicate, i.e., the member partition relation predicate. (Read: 𝐴 is a member partition.)
wff MembPart 𝐴
Syntax	cpetparts 38738	Extend the definition of a class to include the blocklift-stable partitions class.
class PetParts
Syntax	cpet2parts 38739	Extend the definition of a class to include the grade- and blocklift-stable partitions class.
class Pet2Parts
21.26.2  Preparatory theorems
Theorem	el2v1 38740	New way (elv 3462, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 23-Oct-2018.)
⊢ ((𝑥 ∈ V ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	el3v1 38741	New way (elv 3462, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
⊢ ((𝑥 ∈ V ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
Theorem	el3v2 38742	New way (elv 3462, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
⊢ ((𝜑 ∧ 𝑦 ∈ V ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
Theorem	el3v12 38743	New way (elv 3462, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	el3v13 38744	New way (elv 3462, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ ((𝑥 ∈ V ∧ 𝜓 ∧ 𝑧 ∈ V) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	el3v23 38745	New way (elv 3462, 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	anan 38746	Multiple commutations in conjunction. (Contributed by Peter Mazsa, 7-Mar-2020.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜑 ∧ 𝜃) ∧ 𝜏)) ↔ ((𝜓 ∧ 𝜃) ∧ (𝜑 ∧ (𝜒 ∧ 𝜏))))
Theorem	triantru3 38747	A wff is equivalent to its conjunctions with truths. (Contributed by Peter Mazsa, 30-Nov-2018.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ (𝜒 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	biorfd 38748	A wff is equivalent to its disjunction with falsehood, deduction form. (Contributed by Peter Mazsa, 22-Aug-2023.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → (𝜒 ↔ (𝜓 ∨ 𝜒)))
Theorem	eqbrtr 38749	Substitution of equal classes in binary relation. (Contributed by Peter Mazsa, 14-Jun-2024.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Theorem	eqbrb 38750	Substitution of equal classes in a binary relation. (Contributed by Peter Mazsa, 14-Jun-2024.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵𝑅𝐶))
Theorem	eqeltr 38751	Substitution of equal classes into element relation. (Contributed by Peter Mazsa, 22-Jul-2017.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)
Theorem	eqelb 38752	Substitution of equal classes into element relation. (Contributed by Peter Mazsa, 17-Jul-2019.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶))
Theorem	eqeqan2d 38753	Implication of introducing a new equality. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	disjresin 38754	The restriction to a disjoint is the empty class. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = ∅)
Theorem	disjresdisj 38755	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 38756	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 38757	Lemma for ressucdifsn2 38998. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
Theorem	inres2 38758	Two ways of expressing the restriction of an intersection. (Contributed by Peter Mazsa, 5-Jun-2021.)
⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑅 ∩ 𝑆) ↾ 𝐴)
Theorem	coideq 38759	Equality theorem for composition of two classes. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐵))
Theorem	nexmo1 38760	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	eqab2 38761	Implication of a class abstraction. (Contributed by Peter Mazsa, 16-Apr-2019.)
⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑) → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	r2alan 38762*	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 38763	Inference of restricted abstraction subclass from implication. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabimbieq 38764	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 38765*	Intersection with class abstraction. (Contributed by Peter Mazsa, 21-Jul-2021.)
⊢ 𝐴 = (𝐵 ∩ 𝐶)    &   ⊢ 𝐵 = {𝑥 ∣ 𝜑}    ⇒   ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑}
Theorem	abeqinbi 38766*	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	eqrabi 38767*	Class element of a restricted class abstraction. (Contributed by Peter Mazsa, 24-Jul-2021.)
⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))    ⇒   ⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑}
Theorem	rabeqel 38768*	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 38769*	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 38770	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 38771	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 38772*	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	xpv 38773*	Cartesian product of a class and the universe. (Contributed by Peter Mazsa, 6-Oct-2020.)
⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴}
Theorem	vxp 38774*	Cartesian product of the universe and a class. (Contributed by Peter Mazsa, 3-Dec-2020.)
⊢ (V × 𝐴) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝐴}
Theorem	opelvvdif 38775	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 38776*	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 38777	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 38778	Binary relation with universal complement. (Contributed by Peter Mazsa, 14-Jul-2018.)
⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
Theorem	brvvdif 38779	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 38780	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 38781	The converse of the binary epsilon relation. (Contributed by Peter Mazsa, 30-Jan-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝐵 ↔ 𝐵 ∈ 𝐴))
Theorem	elecALTV 38782	Elementhood in the 𝑅-coset of 𝐴. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 8727 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 38783	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 38784	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 38785	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	elec1cnvres 38786	Elementhood in the converse restricted coset of 𝐵. (Contributed by Peter Mazsa, 21-Sep-2018.)
⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ [𝐵]◡(𝑅 ↾ 𝐴) ↔ (𝐶 ∈ 𝐴 ∧ 𝐶𝑅𝐵)))
Theorem	ec1cnvres 38787*	Converse restricted coset of 𝐵. (Contributed by Peter Mazsa, 22-Mar-2019.) (Revised by Peter Mazsa, 21-Oct-2021.)
⊢ (𝐵 ∈ 𝑉 → [𝐵]◡(𝑅 ↾ 𝐴) = {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵})
Theorem	eldmres 38788*	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 38789*	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 38790	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 38791	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 38792*	Elementhood in a domain. (Contributed by Peter Mazsa, 24-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅))
Theorem	eldmres2 38793*	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	eldmres3 38794	Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 23-Nov-2025.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ [𝐵]𝑅 ≠ ∅)))
Theorem	eceq1i 38795	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	ecres 38796*	Restricted coset of 𝐵. (Contributed by Peter Mazsa, 9-Dec-2018.)
⊢ [𝐵](𝑅 ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)}
Theorem	eccnvepres 38797*	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 38798	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 38799	The converse epsilon coset of a set is the set. (Contributed by Peter Mazsa, 27-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)
Theorem	extep 38800	Property of epsilon relation, see also extid 38827, extssr 39100 and the comment of df-ssr 39089. (Contributed by Peter Mazsa, 10-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ E = [𝐵]◡ E ↔ 𝐴 = 𝐵))