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Theorem List for Intuitionistic Logic Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnnm00 6001 The product of two natural numbers is zero iff at least one of them is zero. (Contributed by Jim Kingdon, 11-Nov-2004.)
((A 𝜔 B 𝜔) → ((A ·𝑜 B) = ∅ ↔ (A = ∅ B = ∅)))

2.6.24  Equivalence relations and classes

Syntaxwer 6002 Extend the definition of a wff to include the equivalence predicate.
wff 𝑅 Er A

Syntaxcec 6003 Extend the definition of a class to include equivalence class.
class [A]𝑅

Syntaxcqs 6004 Extend the definition of a class to include quotient set.
class (A / 𝑅)

Definitiondf-er 6005 Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 6006 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6025, ersymb 6019, and ertr 6020. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)
(𝑅 Er A ↔ (Rel 𝑅 dom 𝑅 = A (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))

Theoremdfer2 6006* Alternate definition of equivalence predicate. (Contributed by NM, 3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A ↔ (Rel 𝑅 dom 𝑅 = A xyz((x𝑅yy𝑅x) ((x𝑅y y𝑅z) → x𝑅z))))

Definitiondf-ec 6007 Define the 𝑅-coset of A. Exercise 35 of [Enderton] p. 61. This is called the equivalence class of A modulo 𝑅 when 𝑅 is an equivalence relation (i.e. when Er 𝑅; see dfer2 6006). In this case, A is a representative (member) of the equivalence class [A]𝑅, which contains all sets that are equivalent to A. Definition of [Enderton] p. 57 uses the notation [A] (subscript) 𝑅, although we simply follow the brackets by 𝑅 since we don't have subscripted expressions. For an alternate definition, see dfec2 6008. (Contributed by NM, 23-Jul-1995.)
[A]𝑅 = (𝑅 “ {A})

Theoremdfec2 6008* Alternate definition of 𝑅-coset of A. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
(A 𝑉 → [A]𝑅 = {yA𝑅y})

Theoremecexg 6009 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
(𝑅 B → [A]𝑅 V)

Theoremecexr 6010 An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
(A [B]𝑅B V)

Definitiondf-qs 6011* Define quotient set. 𝑅 is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)
(A / 𝑅) = {yx A y = [x]𝑅}

Theoremereq1 6012 Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝑅 = 𝑆 → (𝑅 Er A𝑆 Er A))

Theoremereq2 6013 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
(A = B → (𝑅 Er A𝑅 Er B))

Theoremerrel 6014 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A → Rel 𝑅)

Theoremerdm 6015 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A → dom 𝑅 = A)

Theoremercl 6016 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)       (φA 𝑋)

Theoremersym 6017 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)       (φB𝑅A)

Theoremercl2 6018 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)       (φB 𝑋)

Theoremersymb 6019 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)       (φ → (A𝑅BB𝑅A))

Theoremertr 6020 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)       (φ → ((A𝑅B B𝑅𝐶) → A𝑅𝐶))

Theoremertrd 6021 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)    &   (φB𝑅𝐶)       (φA𝑅𝐶)

Theoremertr2d 6022 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)    &   (φB𝑅𝐶)       (φ𝐶𝑅A)

Theoremertr3d 6023 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(φ𝑅 Er 𝑋)    &   (φB𝑅A)    &   (φB𝑅𝐶)       (φA𝑅𝐶)

Theoremertr4d 6024 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)    &   (φ𝐶𝑅B)       (φA𝑅𝐶)

Theoremerref 6025 An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)    &   (φA 𝑋)       (φA𝑅A)

Theoremercnv 6026 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A𝑅 = 𝑅)

Theoremerrn 6027 The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A → ran 𝑅 = A)

Theoremerssxp 6028 An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A𝑅 ⊆ (A × A))

Theoremerex 6029 An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A → (A 𝑉𝑅 V))

Theoremerexb 6030 An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A → (𝑅 V ↔ A V))

Theoremiserd 6031* A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ → Rel 𝑅)    &   ((φ x𝑅y) → y𝑅x)    &   ((φ (x𝑅y y𝑅z)) → x𝑅z)    &   (φ → (x Ax𝑅x))       (φ𝑅 Er A)

Theorembrdifun 6032 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))       ((A 𝑋 B 𝑋) → (A𝑅B ↔ ¬ (A < B B < A)))

Theoremswoer 6033* Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))    &   ((φ (y 𝑋 z 𝑋)) → (y < z → ¬ z < y))    &   ((φ (x 𝑋 y 𝑋 z 𝑋)) → (x < y → (x < z z < y)))       (φ𝑅 Er 𝑋)

Theoremswoord1 6034* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))    &   ((φ (y 𝑋 z 𝑋)) → (y < z → ¬ z < y))    &   ((φ (x 𝑋 y 𝑋 z 𝑋)) → (x < y → (x < z z < y)))    &   (φB 𝑋)    &   (φ𝐶 𝑋)    &   (φA𝑅B)       (φ → (A < 𝐶B < 𝐶))

Theoremswoord2 6035* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))    &   ((φ (y 𝑋 z 𝑋)) → (y < z → ¬ z < y))    &   ((φ (x 𝑋 y 𝑋 z 𝑋)) → (x < y → (x < z z < y)))    &   (φB 𝑋)    &   (φ𝐶 𝑋)    &   (φA𝑅B)       (φ → (𝐶 < A𝐶 < B))

Theoremeqerlem 6036* Lemma for eqer 6037. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
(x = yA = B)    &   𝑅 = {⟨x, y⟩ ∣ A = B}       (z𝑅wz / xA = w / xA)

Theoremeqer 6037* Equivalence relation involving equality of dependent classes A(x) and B(y). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)
(x = yA = B)    &   𝑅 = {⟨x, y⟩ ∣ A = B}       𝑅 Er V

Theoremider 6038 The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
I Er V

Theorem0er 6039 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
∅ Er ∅

Theoremeceq1 6040 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
(A = B → [A]𝐶 = [B]𝐶)

Theoremeceq1d 6041 Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
(φA = B)       (φ → [A]𝐶 = [B]𝐶)

Theoremeceq2 6042 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
(A = B → [𝐶]A = [𝐶]B)

Theoremelecg 6043 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
((A 𝑉 B 𝑊) → (A [B]𝑅B𝑅A))

Theoremelec 6044 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)
A V    &   B V       (A [B]𝑅B𝑅A)

Theoremrelelec 6045 Membership in an equivalence class when 𝑅 is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
(Rel 𝑅 → (A [B]𝑅B𝑅A))

Theoremecss 6046 An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)       (φ → [A]𝑅𝑋)

Theoremecdmn0m 6047* A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)
(A dom 𝑅x x [A]𝑅)

Theoremereldm 6048 Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)    &   (φ → [A]𝑅 = [B]𝑅)       (φ → (A 𝑋B 𝑋))

Theoremerth 6049 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
(φ𝑅 Er 𝑋)    &   (φA 𝑋)       (φ → (A𝑅B ↔ [A]𝑅 = [B]𝑅))

Theoremerth2 6050 Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
(φ𝑅 Er 𝑋)    &   (φB 𝑋)       (φ → (A𝑅B ↔ [A]𝑅 = [B]𝑅))

Theoremerthi 6051 Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)       (φ → [A]𝑅 = [B]𝑅)

Theoremecidsn 6052 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
[A] I = {A}

Theoremqseq1 6053 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
(A = B → (A / 𝐶) = (B / 𝐶))

Theoremqseq2 6054 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
(A = B → (𝐶 / A) = (𝐶 / B))

Theoremelqsg 6055* Closed form of elqs 6056. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(B 𝑉 → (B (A / 𝑅) ↔ x A B = [x]𝑅))

Theoremelqs 6056* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
B V       (B (A / 𝑅) ↔ x A B = [x]𝑅)

Theoremelqsi 6057* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
(B (A / 𝑅) → x A B = [x]𝑅)

Theoremecelqsg 6058 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
((𝑅 𝑉 B A) → [B]𝑅 (A / 𝑅))

Theoremecelqsi 6059 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝑅 V       (B A → [B]𝑅 (A / 𝑅))

Theoremecopqsi 6060 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)
𝑅 V    &   𝑆 = ((A × A) / 𝑅)       ((B A 𝐶 A) → [⟨B, 𝐶⟩]𝑅 𝑆)

Theoremqsexg 6061 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
(A 𝑉 → (A / 𝑅) V)

Theoremqsex 6062 A quotient set exists. (Contributed by NM, 14-Aug-1995.)
A V       (A / 𝑅) V

Theoremuniqs 6063 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
(𝑅 𝑉 (A / 𝑅) = (𝑅A))

Theoremqsss 6064 A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er A)       (φ → (A / 𝑅) ⊆ 𝒫 A)

Theoremuniqs2 6065 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
(φ𝑅 Er A)    &   (φ𝑅 𝑉)       (φ (A / 𝑅) = A)

Theoremsnec 6066 The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)
A V       {[A]𝑅} = ({A} / 𝑅)

Theoremecqs 6067 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
𝑅 V       [A]𝑅 = ({A} / 𝑅)

Theoremecid 6068 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
A V       [A] E = A

Theoremecidg 6069 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.)
(A 𝑉 → [A] E = A)

Theoremqsid 6070 A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
(A / E ) = A

Theoremectocld 6071* Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)
𝑆 = (B / 𝑅)    &   ([x]𝑅 = A → (φψ))    &   ((χ x B) → φ)       ((χ A 𝑆) → ψ)

Theoremectocl 6072* Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝑆 = (B / 𝑅)    &   ([x]𝑅 = A → (φψ))    &   (x Bφ)       (A 𝑆ψ)

Theoremelqsn0m 6073* An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)
((dom 𝑅 = A B (A / 𝑅)) → x x B)

Theoremelqsn0 6074 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
((dom 𝑅 = A B (A / 𝑅)) → B ≠ ∅)

Theoremecelqsdm 6075 Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)
((dom 𝑅 = A [B]𝑅 (A / 𝑅)) → B A)

Theoremxpiderm 6076* A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by Jim Kingdon, 22-Aug-2019.)
(x x A → (A × A) Er A)

Theoremiinerm 6077* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
((y y A x A 𝑅 Er B) → x A 𝑅 Er B)

Theoremriinerm 6078* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
((y y A x A 𝑅 Er B) → ((B × B) ∩ x A 𝑅) Er B)

Theoremerinxp 6079 A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er A)    &   (φBA)       (φ → (𝑅 ∩ (B × B)) Er B)

Theoremecinxp 6080 Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)
(((𝑅A) ⊆ A B A) → [B]𝑅 = [B](𝑅 ∩ (A × A)))

Theoremqsinxp 6081 Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
((𝑅A) ⊆ A → (A / 𝑅) = (A / (𝑅 ∩ (A × A))))

Theoremqsel 6082 If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.)
((𝑅 Er 𝑋 B (A / 𝑅) 𝐶 B) → B = [𝐶]𝑅)

Theoremqliftlem 6083* 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)    &   ((φ x 𝑋) → A 𝑌)    &   (φ𝑅 Er 𝑋)    &   (φ𝑋 V)       ((φ x 𝑋) → [x]𝑅 (𝑋 / 𝑅))

Theoremqliftrel 6084* 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)    &   ((φ x 𝑋) → A 𝑌)    &   (φ𝑅 Er 𝑋)    &   (φ𝑋 V)       (φ𝐹 ⊆ ((𝑋 / 𝑅) × 𝑌))

Theoremqliftel 6085* Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)    &   ((φ x 𝑋) → A 𝑌)    &   (φ𝑅 Er 𝑋)    &   (φ𝑋 V)       (φ → ([𝐶]𝑅𝐹𝐷x 𝑋 (𝐶𝑅x 𝐷 = A)))

Theoremqliftel1 6086* Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)    &   ((φ x 𝑋) → A 𝑌)    &   (φ𝑅 Er 𝑋)    &   (φ𝑋 V)       ((φ x 𝑋) → [x]𝑅𝐹A)

Theoremqliftfun 6087* The function 𝐹 is the unique function defined by 𝐹‘[x] = A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)    &   ((φ x 𝑋) → A 𝑌)    &   (φ𝑅 Er 𝑋)    &   (φ𝑋 V)    &   (x = yA = B)       (φ → (Fun 𝐹xy(x𝑅yA = B)))

Theoremqliftfund 6088* The function 𝐹 is the unique function defined by 𝐹‘[x] = A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)    &   ((φ x 𝑋) → A 𝑌)    &   (φ𝑅 Er 𝑋)    &   (φ𝑋 V)    &   (x = yA = B)    &   ((φ x𝑅y) → A = B)       (φ → Fun 𝐹)

Theoremqliftfuns 6089* The function 𝐹 is the unique function defined by 𝐹‘[x] = A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)    &   ((φ x 𝑋) → A 𝑌)    &   (φ𝑅 Er 𝑋)    &   (φ𝑋 V)       (φ → (Fun 𝐹yz(y𝑅zy / xA = z / xA)))

Theoremqliftf 6090* The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)    &   ((φ x 𝑋) → A 𝑌)    &   (φ𝑅 Er 𝑋)    &   (φ𝑋 V)       (φ → (Fun 𝐹𝐹:(𝑋 / 𝑅)⟶𝑌))

Theoremqliftval 6091* The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)    &   ((φ x 𝑋) → A 𝑌)    &   (φ𝑅 Er 𝑋)    &   (φ𝑋 V)    &   (x = 𝐶A = B)    &   (φ → Fun 𝐹)       ((φ 𝐶 𝑋) → (𝐹‘[𝐶]𝑅) = B)

Theoremecoptocl 6092* Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)
𝑆 = ((B × 𝐶) / 𝑅)    &   ([⟨x, y⟩]𝑅 = A → (φψ))    &   ((x B y 𝐶) → φ)       (A 𝑆ψ)

Theorem2ecoptocl 6093* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)
𝑆 = ((𝐶 × 𝐷) / 𝑅)    &   ([⟨x, y⟩]𝑅 = A → (φψ))    &   ([⟨z, w⟩]𝑅 = B → (ψχ))    &   (((x 𝐶 y 𝐷) (z 𝐶 w 𝐷)) → φ)       ((A 𝑆 B 𝑆) → χ)

Theorem3ecoptocl 6094* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)
𝑆 = ((𝐷 × 𝐷) / 𝑅)    &   ([⟨x, y⟩]𝑅 = A → (φψ))    &   ([⟨z, w⟩]𝑅 = B → (ψχ))    &   ([⟨v, u⟩]𝑅 = 𝐶 → (χθ))    &   (((x 𝐷 y 𝐷) (z 𝐷 w 𝐷) (v 𝐷 u 𝐷)) → φ)       ((A 𝑆 B 𝑆 𝐶 𝑆) → θ)

Theorembrecop 6095* Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)
V    &    Er (𝐺 × 𝐺)    &   𝐻 = ((𝐺 × 𝐺) / )    &    = {⟨x, y⟩ ∣ ((x 𝐻 y 𝐻) zwvu((x = [⟨z, w⟩] y = [⟨v, u⟩] ) φ))}    &   ((((z 𝐺 w 𝐺) (A 𝐺 B 𝐺)) ((v 𝐺 u 𝐺) (𝐶 𝐺 𝐷 𝐺))) → (([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ) → (φψ)))       (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → ([⟨A, B⟩] [⟨𝐶, 𝐷⟩] ψ))

Theoremeroveu 6096* Lemma for eroprf 6098. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝐽 = (A / 𝑅)    &   𝐾 = (B / 𝑆)    &   (φ𝑇 𝑍)    &   (φ𝑅 Er 𝑈)    &   (φ𝑆 Er 𝑉)    &   (φ𝑇 Er 𝑊)    &   (φA𝑈)    &   (φB𝑉)    &   (φ𝐶𝑊)    &   (φ+ :(A × B)⟶𝐶)    &   ((φ ((𝑟 A 𝑠 A) (𝑡 B u B))) → ((𝑟𝑅𝑠 𝑡𝑆u) → (𝑟 + 𝑡)𝑇(𝑠 + u)))       ((φ (𝑋 𝐽 𝑌 𝐾)) → ∃!z𝑝 A 𝑞 B ((𝑋 = [𝑝]𝑅 𝑌 = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))

Theoremerovlem 6097* Lemma for eroprf 6098. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
𝐽 = (A / 𝑅)    &   𝐾 = (B / 𝑆)    &   (φ𝑇 𝑍)    &   (φ𝑅 Er 𝑈)    &   (φ𝑆 Er 𝑉)    &   (φ𝑇 Er 𝑊)    &   (φA𝑈)    &   (φB𝑉)    &   (φ𝐶𝑊)    &   (φ+ :(A × B)⟶𝐶)    &   ((φ ((𝑟 A 𝑠 A) (𝑡 B u B))) → ((𝑟𝑅𝑠 𝑡𝑆u) → (𝑟 + 𝑡)𝑇(𝑠 + u)))    &    = {⟨⟨x, y⟩, z⟩ ∣ 𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)}       (φ = (x 𝐽, y 𝐾 ↦ (℩z𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇))))

Theoremeroprf 6098* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
𝐽 = (A / 𝑅)    &   𝐾 = (B / 𝑆)    &   (φ𝑇 𝑍)    &   (φ𝑅 Er 𝑈)    &   (φ𝑆 Er 𝑉)    &   (φ𝑇 Er 𝑊)    &   (φA𝑈)    &   (φB𝑉)    &   (φ𝐶𝑊)    &   (φ+ :(A × B)⟶𝐶)    &   ((φ ((𝑟 A 𝑠 A) (𝑡 B u B))) → ((𝑟𝑅𝑠 𝑡𝑆u) → (𝑟 + 𝑡)𝑇(𝑠 + u)))    &    = {⟨⟨x, y⟩, z⟩ ∣ 𝑝 A 𝑞 B ((x = [𝑝]𝑅 y = [𝑞]𝑆) z = [(𝑝 + 𝑞)]𝑇)}    &   (φ𝑅 𝑋)    &   (φ𝑆 𝑌)    &   𝐿 = (𝐶 / 𝑇)       (φ :(𝐽 × 𝐾)⟶𝐿)

Theoremeroprf2 6099* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐽 = (A / )    &    = {⟨⟨x, y⟩, z⟩ ∣ 𝑝 A 𝑞 A ((x = [𝑝] y = [𝑞] ) z = [(𝑝 + 𝑞)] )}    &   (φ 𝑋)    &   (φ Er 𝑈)    &   (φA𝑈)    &   (φ+ :(A × A)⟶A)    &   ((φ ((𝑟 A 𝑠 A) (𝑡 A u A))) → ((𝑟 𝑠 𝑡 u) → (𝑟 + 𝑡) (𝑠 + u)))       (φ :(𝐽 × 𝐽)⟶𝐽)

Theoremecopoveq 6100* This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation (specified by the hypothesis) in terms of its operation 𝐹. (Contributed by NM, 16-Aug-1995.)
= {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z + u) = (w + v)))}       (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → (⟨A, B𝐶, 𝐷⟩ ↔ (A + 𝐷) = (B + 𝐶)))

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