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Theorem List for Metamath Proof Explorer - 34101-34200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmpt2bi123f 34101* Equality deduction for maps-to notations with two arguments. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵    &   𝑦𝐴    &   𝑦𝐵    &   𝑦𝐶    &   𝑦𝐷    &   𝑥𝐶    &   𝑥𝐷       (((𝐴 = 𝐵𝐶 = 𝐷) ∧ ∀𝑥𝐴𝑦𝐶 𝐸 = 𝐹) → (𝑥𝐴, 𝑦𝐶𝐸) = (𝑥𝐵, 𝑦𝐷𝐹))

Theoremiuneq12f 34102 Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremiineq12f 34103 Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremopabbi 34104 Equality deduction for class abstraction of ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
(∀𝑥𝑦(𝜑𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})

Theoremmptbi12f 34105 Equality deduction for maps-to notations. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐷 = 𝐸) → (𝑥𝐴𝐷) = (𝑥𝐵𝐸))

20.20.4  Miscellanea

Work in progress or things that do not belong anywhere else.

Theoremscottexf 34106* A version of scottex 8786 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
𝑦𝐴    &   𝑥𝐴       {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V

Theoremscott0f 34107* A version of scott0 8787 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
𝑦𝐴    &   𝑥𝐴       (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)

Theoremscottn0f 34108* A version of scott0f 34107 with inequalities instead of equalities. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
𝑦𝐴    &   𝑥𝐴       (𝐴 ≠ ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)

Theoremac6s3f 34109* Generalization of the Axiom of Choice to classes, with bound-variable hypothesis. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝜑 → ∃𝑓𝑥𝐴 𝜓)

Theoremac6s6 34110* Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       𝑓𝑥𝐴 (∃𝑦𝜑𝜓)

Theoremac6s6f 34111* Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 20-Aug-2018.)
𝐴 ∈ V    &   𝑦𝜓    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))    &   𝑥𝐴       𝑓𝑥𝐴 (∃𝑦𝜑𝜓)

20.21  Mathbox for Peter Mazsa

20.21.1  Notations

Syntaxcxrn 34112 Extend the definition of a class to include the range Cartesian product class.
class (𝐴𝐵)

Syntaxccoss 34113 Extend the definition of a class to include the class of cosets by a class. (Read: the class of cosets by 𝑅.)
class 𝑅

Syntaxccoels 34114 Extend the definition of a class to include the class of coelements on a class. (Read: the class of coelements on 𝐴.)
class 𝐴

Syntaxcrels 34115 Extend the definition of a class to include the relation class.
class Rels

Syntaxcssr 34116 Extend the definition of a class to include the subset class.
class S

Syntaxcrefs 34117 Extend the definition of a class to include the reflexivity class.
class Refs

Syntaxcrefrels 34118 Extend the definition of a class to include the reflexive relations class.
class RefRels

Syntaxwrefrel 34119 Extend the definition of a wff to include the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.)
wff RefRel 𝑅

Syntaxccnvrefs 34120 Extend the definition of a class to include the converse reflexivity class.
class CnvRefs

Syntaxccnvrefrels 34121 Extend the definition of a class to include the converse reflexive relations class.
class CnvRefRels

Syntaxwcnvrefrel 34122 Extend the definition of a wff to include the converse reflexive relation predicate. (Read: 𝑅 is a converse reflexive relation.)
wff CnvRefRel 𝑅

Syntaxcsyms 34123 Extend the definition of a class to include the symmetry class.
class Syms

Syntaxcsymrels 34124 Extend the definition of a class to include the symmetry relations class.
class SymRels

Syntaxwsymrel 34125 Extend the definition of a wff to include the symmetry relation predicate. (Read: 𝑅 is a symmetric relation.)
wff SymRel 𝑅

20.21.2  Preparatory theorems

Theoremelv 34126 New way (elv 34126, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. Inference forms (with 𝐴 ∈ V hypotheses) of the general theorems (proving (𝐴𝑉 assertions) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.)
(𝑥 ∈ V → 𝜑)       𝜑

Theoremel2v 34127 New way (elv 34126, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. Inference forms (with 𝐴 ∈ V and 𝐵 ∈ V hypotheses) of the general theorems (proving ((𝐴𝑉𝐵𝑊) → assertions) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.)
((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑)       𝜑

Theoremel2v1 34128 New way (elv 34126, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 23-Oct-2018.)
((𝑥 ∈ V ∧ 𝜑) → 𝜓)       (𝜑𝜓)

Theoremel2v2 34129 New way (elv 34126, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 23-Oct-2018.)
((𝜑𝑦 ∈ V) → 𝜓)       (𝜑𝜓)

Theoremel3v 34130 New way (elv 34126, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. Inference forms (with 𝐴 ∈ V, 𝐵 ∈ V and 𝐶 ∈ V hypotheses) of the general theorems (proving ((𝐴𝑉𝐵𝑊𝐶𝑋) → assertions) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.)
((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑)       𝜑

Theoremel3v1 34131 New way (elv 34126, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
((𝑥 ∈ V ∧ 𝜓𝜒) → 𝜃)       ((𝜓𝜒) → 𝜃)

Theoremel3v2 34132 New way (elv 34126, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
((𝜑𝑦 ∈ V ∧ 𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)

Theoremel3v3 34133 New way (elv 34126, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
((𝜑𝜓𝑧 ∈ V) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremel3v12 34134 New way (elv 34126, 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)       (𝜒𝜃)

Theoremel3v13 34135 New way (elv 34126, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
((𝑥 ∈ V ∧ 𝜓𝑧 ∈ V) → 𝜃)       (𝜓𝜃)

Theoremel3v23 34136 New way (elv 34126, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
((𝜑𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜃)       (𝜑𝜃)

Theorembiancom 34137 Commuting conjunction in a biconditional. (Contributed by Peter Mazsa, 17-Jun-2018.)
(𝜑 ↔ (𝜒𝜓))       (𝜑 ↔ (𝜓𝜒))

Theorembiancomd 34138 Commuting conjunction in a biconditional, deduction form. (Contributed by Peter Mazsa, 3-Oct-2018.)
(𝜑 → (𝜓 ↔ (𝜃𝜒)))       (𝜑 → (𝜓 ↔ (𝜒𝜃)))

Theoremanbi1ci 34139 Introduce a left and the same right conjunct to the sides of a logical equivalence. (Contributed by Peter Mazsa, 7-Mar-2020.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜓𝜒))

Theoremanbi1cd 34140 Introduce a left and the same right conjunct to the sides of a logical equivalence, deduction form. (Contributed by Peter Mazsa, 22-May-2021.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜒𝜃)))

Theoreman2anr 34141 Double commutation in conjunction. (Contributed by Peter Mazsa, 27-Jun-2019.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜓𝜑) ∧ (𝜃𝜒)))

Theoremanan 34142 Multiple commutations in conjunction. (Contributed by Peter Mazsa, 7-Mar-2020.)
((((𝜑𝜓) ∧ 𝜒) ∧ ((𝜑𝜃) ∧ 𝜏)) ↔ ((𝜓𝜃) ∧ (𝜑 ∧ (𝜒𝜏))))

Theoremtriantru3 34143 A wff is equivalent to its conjunctions with truths. (Contributed by Peter Mazsa, 30-Nov-2018.)
𝜑    &   𝜓       (𝜒 ↔ (𝜑𝜓𝜒))

Theoremeqeltr 34144 Substitution of equal classes into elementhood relation. (Contributed by Peter Mazsa, 22-Jul-2017.)
((𝐴 = 𝐵𝐵𝐶) → 𝐴𝐶)

Theoremeqelb 34145 Substitution of equal classes into elementhood relation. (Contributed by Peter Mazsa, 17-Jul-2019.)
((𝐴 = 𝐵𝐴𝐶) ↔ (𝐴 = 𝐵𝐵𝐶))

Theoremeqeqan1d 34146 Implication of introducing a new equality. (Contributed by Peter Mazsa, 17-Apr-2019.)
(𝜑𝐴 = 𝐵)       ((𝜑𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremeqeqan2d 34147 Implication of introducing a new equality. (Contributed by Peter Mazsa, 17-Apr-2019.)
(𝜑𝐶 = 𝐷)       ((𝐴 = 𝐵𝜑) → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremineqcom 34148 Two ways of saying that two classes are disjoint (when 𝐶 = ∅: ((𝐴𝐵) = ∅ ↔ (𝐵𝐴) = ∅)). (Contributed by Peter Mazsa, 22-Mar-2017.)
((𝐴𝐵) = 𝐶 ↔ (𝐵𝐴) = 𝐶)

Theoremineqcomi 34149 Disjointness inference (when 𝐶 = ∅), inference form of ineqcom 34148. (Contributed by Peter Mazsa, 26-Mar-2017.)
(𝐴𝐵) = 𝐶       (𝐵𝐴) = 𝐶

Theoreminres2 34150 Two ways of expressing the restriction of an intersection. (Contributed by Peter Mazsa, 5-Jun-2021.)
((𝑅𝐴) ∩ 𝑆) = ((𝑅𝑆) ↾ 𝐴)

Theoremcoideq 34151 Equality theorem for composition of two classes. (Contributed by Peter Mazsa, 23-Sep-2021.)
(𝐴 = 𝐵 → (𝐴𝐴) = (𝐵𝐵))

Theoremralanid 34152 Cancellation law for restriction. (Contributed by Peter Mazsa, 30-Dec-2018.)
(∀𝑥𝐴 (𝑥𝐴𝜑) ↔ ∀𝑥𝐴 𝜑)

Theoremnexmo 34153 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)

Theorem3albii 34154 Inference adding three universal quantifiers to both sides of an equivalence. (Contributed by Peter Mazsa, 10-Aug-2018.)
(𝜑𝜓)       (∀𝑥𝑦𝑧𝜑 ↔ ∀𝑥𝑦𝑧𝜓)

Theorem3ralbii 34155 Inference adding three restricted universal quantifiers to both sides of an equivalence. (Contributed by Peter Mazsa, 25-Jul-2019.)
(𝜑𝜓)       (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜓)

Theoremrabbieq 34156 Equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 8-Jul-2019.)
𝐵 = {𝑥𝐴𝜑}    &   (𝜑𝜓)       𝐵 = {𝑥𝐴𝜓}

Theoremrabimbieq 34157 Restricted equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 22-Jul-2021.)
𝐵 = {𝑥𝐴𝜑}    &   (𝑥𝐴 → (𝜑𝜓))       𝐵 = {𝑥𝐴𝜓}

Theoremabeqin 34158* Intersection with class abstraction. (Contributed by Peter Mazsa, 21-Jul-2021.)
𝐴 = (𝐵𝐶)    &   𝐵 = {𝑥𝜑}       𝐴 = {𝑥𝐶𝜑}

Theoremabeqinbi 34159* Intersection with class abstraction and equivalent wff's. (Contributed by Peter Mazsa, 21-Jul-2021.)
𝐴 = (𝐵𝐶)    &   𝐵 = {𝑥𝜑}    &   (𝑥𝐶 → (𝜑𝜓))       𝐴 = {𝑥𝐶𝜓}

Theoremrabeqel 34160* Class element of a restricted class abstraction. (Contributed by Peter Mazsa, 24-Jul-2021.)
𝐵 = {𝑥𝐴𝜑}    &   (𝑥 = 𝐶 → (𝜑𝜓))       (𝐶𝐵 ↔ (𝜓𝐶𝐴))

Theoremeqrelf 34161* 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 34162 Elementhood in a converse 𝑅-coset when 𝑅 is a relation. (Contributed by Peter Mazsa, 9-Dec-2018.)
(Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐴𝑅𝐵))

Theoremreleccnveq 34163* 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 34164 Negated elementhood of ordered pair. (Contributed by Peter Mazsa, 14-Jan-2019.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ ((V × V) ∖ 𝑅) ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅))

Theoremvvdifopab 34165* Ordered-pair class abstraction defined by a negation. (Contributed by Peter Mazsa, 25-Jun-2019.)
((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}

Theorembrvdif 34166 Binary relation with universal complement is the negation of the relation. (Contributed by Peter Mazsa, 1-Jul-2018.)
(𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)

Theorembrvdif2 34167 Binary relation with universal complement. (Contributed by Peter Mazsa, 14-Jul-2018.)
(𝐴(V ∖ 𝑅)𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅)

Theorembrvvdif 34168 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 34169 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 34170 The converse of the binary epsilon relation. (Contributed by Peter Mazsa, 30-Jan-2018.)
(𝐴𝑉 → (𝐴 E 𝐵𝐵𝐴))

TheoremelecALTV 34171 Elementhood in the 𝑅-coset of 𝐴. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 7828 with this original form of Suppes. Peter Mazsa) (Contributed by Mario Carneiro, 9-Jul-2014.)
((𝐴𝑉𝐵𝑊) → (𝐵 ∈ [𝐴]𝑅𝐴𝑅𝐵))

TheoremopelresALTV 34172 Ordered pair elementhood in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
(𝐶𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))

TheorembrresALTV 34173 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
(𝐶𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))

Theorembrcnvepres 34174 Restricted converse epsilon binary relation. (Contributed by Peter Mazsa, 10-Feb-2018.)
((𝐵𝑉𝐶𝑊) → (𝐵( E ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐶𝐵)))

Theorembrinxp2ALTV 34175 Intersection with cross product binary relation . (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶𝐴𝐷𝐵) ∧ 𝐶𝑅𝐷))

Theorembrres2 34176 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 34177* Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 9-Jan-2019.)
(𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦)))

Theoremeldm4 34178* Elementhood in a domain. (Contributed by Peter Mazsa, 24-Oct-2018.)
(𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅))

Theoremeldmres2 34179* Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 21-Aug-2020.)
(𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ (𝐵𝐴 ∧ ∃𝑦 𝑦 ∈ [𝐵]𝑅)))

Theoremeceq1i 34180 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

Theoremeceq2i 34181 Equality theorem for the 𝐴-coset and 𝐵-coset of 𝐶, inference version. (Contributed by Peter Mazsa, 11-May-2021.)
𝐴 = 𝐵       [𝐶]𝐴 = [𝐶]𝐵

Theoremeceq2d 34182 Equality theorem for the 𝐴-coset and 𝐵-coset of 𝐶, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → [𝐶]𝐴 = [𝐶]𝐵)

Theoremelecres 34183 Elementhood in the restricted coset of 𝐵. (Contributed by Peter Mazsa, 21-Sep-2018.)
(𝐶𝑉 → (𝐶 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝐶)))

Theoremecres 34184* Restricted coset of 𝐵. (Contributed by Peter Mazsa, 9-Dec-2018.)
[𝐵](𝑅𝐴) = {𝑥 ∣ (𝐵𝐴𝐵𝑅𝑥)}

Theoremecres2 34185 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 34186* 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 34187 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 34188 The converse epsilon coset of a set is the set. (Contributed by Peter Mazsa, 27-Jan-2019.)
(𝐴𝑉 → [𝐴] E = 𝐴)

Theoremextep 34189 Property of epsilon relation, cf. extid 34222, extssr 34399 and the comment of df-ssr 34388. (Contributed by Peter Mazsa, 10-Jul-2019.)
((𝐴𝑉𝐵𝑊) → ([𝐴] E = [𝐵] E ↔ 𝐴 = 𝐵))

Theoremeccnvepres2 34190 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 34191 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 34192* 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 34193* 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 34194 Subclass theorem for quotient sets. (Contributed by Peter Mazsa, 12-Sep-2020.)
(𝐴𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶))

Theoremqseq1i 34195 Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.)
𝐴 = 𝐵       (𝐴 / 𝐶) = (𝐵 / 𝐶)

Theoremqseq1d 34196 Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶))

Theoremqseq2i 34197 Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.)
𝐴 = 𝐵       (𝐶 / 𝐴) = (𝐶 / 𝐵)

Theoremqseq2d 34198 Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶 / 𝐴) = (𝐶 / 𝐵))

Theoremqseq12 34199 Equality theorem for quotient set. (Contributed by Peter Mazsa, 17-Apr-2019.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Theorembrinxprnres 34200 Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.)
(𝐶𝑉 → (𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))

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