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Theorem List for Metamath Proof Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremerrn 8301 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 8302 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 8303 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 → (𝐴𝑉𝑅 ∈ V))
 
Theoremerexb 8304 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 8305* 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)    &   ((𝜑𝑥𝑅𝑦) → 𝑦𝑅𝑥)    &   ((𝜑 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)    &   (𝜑 → (𝑥𝐴𝑥𝑅𝑥))       (𝜑𝑅 Er 𝐴)
 
Theoremiseri 8306* A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 8305, which avoids the need to provide a "dummy antecedent" 𝜑 if there is no natural one to choose. (Contributed by AV, 30-Apr-2021.)
Rel 𝑅    &   (𝑥𝑅𝑦𝑦𝑅𝑥)    &   ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)    &   (𝑥𝐴𝑥𝑅𝑥)       𝑅 Er 𝐴
 
TheoremiseriALT 8307* Alternate proof of iseri 8306, avoiding the usage of mptru 1535 and as antecedent by using ax-mp 5 and one of the hypotheses as antecedent. This results, however, in a slightly longer proof. (Contributed by AV, 30-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Rel 𝑅    &   (𝑥𝑅𝑦𝑦𝑅𝑥)    &   ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)    &   (𝑥𝐴𝑥𝑅𝑥)       𝑅 Er 𝐴
 
Theorembrdifun 8308 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))       ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
 
Theoremswoer 8309* 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 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))       (𝜑𝑅 Er 𝑋)
 
Theoremswoord1 8310* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))    &   ((𝜑 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑 → (𝐴 < 𝐶𝐵 < 𝐶))
 
Theoremswoord2 8311* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))    &   ((𝜑 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
 
Theoremswoso 8312* If the incomparability relation is equivalent to equality in a subset, then the partial order strictly orders the subset. (Contributed by Mario Carneiro, 30-Dec-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))    &   ((𝜑 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))    &   (𝜑𝑌𝑋)    &   ((𝜑 ∧ (𝑥𝑌𝑦𝑌𝑥𝑅𝑦)) → 𝑥 = 𝑦)       (𝜑< Or 𝑌)
 
Theoremeqerlem 8313* Lemma for eqer 8314. (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 8314* 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.) (Proof shortened by AV, 1-May-2021.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}       𝑅 Er V
 
Theoremider 8315 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 8316 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 1-May-2021.)
∅ Er ∅
 
Theoremeceq1 8317 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
(𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
 
Theoremeceq1d 8318 Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → [𝐴]𝐶 = [𝐵]𝐶)
 
Theoremeceq2 8319 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
(𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
 
Theoremeceq2i 8320 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 8321 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)
 
Theoremelecg 8322 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
 
Theoremelec 8323 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 8324 Membership in an equivalence class when 𝑅 is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
(Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
 
Theoremecss 8325 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)
 
Theoremecdmn0 8326 A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
(𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
 
Theoremereldm 8327 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 → (𝐴𝑋𝐵𝑋))
 
Theoremerth 8328 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)    &   (𝜑𝐴𝑋)       (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
 
Theoremerth2 8329 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)    &   (𝜑𝐵𝑋)       (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
 
Theoremerthi 8330 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)
 
Theoremerdisj 8331 Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83. (Contributed by NM, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.)
(𝑅 Er 𝑋 → ([𝐴]𝑅 = [𝐵]𝑅 ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))
 
Theoremecidsn 8332 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
[𝐴] I = {𝐴}
 
Theoremqseq1 8333 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
(𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶))
 
Theoremqseq2 8334 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
(𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵))
 
Theoremqseq2i 8335 Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.)
𝐴 = 𝐵       (𝐶 / 𝐴) = (𝐶 / 𝐵)
 
Theoremqseq2d 8336 Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶 / 𝐴) = (𝐶 / 𝐵))
 
Theoremqseq12 8337 Equality theorem for quotient set. (Contributed by Peter Mazsa, 17-Apr-2019.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))
 
Theoremelqsg 8338* Closed form of elqs 8339. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝐵𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
 
Theoremelqs 8339* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
𝐵 ∈ V       (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅)
 
Theoremelqsi 8340* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
(𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥𝐴 𝐵 = [𝑥]𝑅)
 
Theoremelqsecl 8341* Membership in a quotient set by an equivalence class according to . (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by AV, 30-Apr-2021.)
(𝐵𝑋 → (𝐵 ∈ (𝑊 / ) ↔ ∃𝑥𝑊 𝐵 = {𝑦𝑥 𝑦}))
 
Theoremecelqsg 8342 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
((𝑅𝑉𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
 
Theoremecelqsi 8343 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝑅 ∈ V       (𝐵𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
 
Theoremecopqsi 8344 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)
𝑅 ∈ V    &   𝑆 = ((𝐴 × 𝐴) / 𝑅)       ((𝐵𝐴𝐶𝐴) → [⟨𝐵, 𝐶⟩]𝑅𝑆)
 
Theoremqsexg 8345 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
(𝐴𝑉 → (𝐴 / 𝑅) ∈ V)
 
Theoremqsex 8346 A quotient set exists. (Contributed by NM, 14-Aug-1995.)
𝐴 ∈ V       (𝐴 / 𝑅) ∈ V
 
Theoremuniqs 8347 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
(𝑅𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
 
Theoremqsss 8348 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 8349 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
(𝜑𝑅 Er 𝐴)    &   (𝜑𝑅𝑉)       (𝜑 (𝐴 / 𝑅) = 𝐴)
 
Theoremsnec 8350 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 8351 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
𝑅 ∈ V       [𝐴]𝑅 = ({𝐴} / 𝑅)
 
Theoremecid 8352 A set is equal to its coset under the converse membership relation. (Note: the converse membership relation is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝐴 ∈ V       [𝐴] E = 𝐴
 
Theoremqsid 8353 A set is equal to its quotient set modulo the converse membership relation. (Note: the converse membership relation is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
(𝐴 / E ) = 𝐴
 
Theoremectocld 8354* Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)
𝑆 = (𝐵 / 𝑅)    &   ([𝑥]𝑅 = 𝐴 → (𝜑𝜓))    &   ((𝜒𝑥𝐵) → 𝜑)       ((𝜒𝐴𝑆) → 𝜓)
 
Theoremectocl 8355* 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))    &   (𝑥𝐵𝜑)       (𝐴𝑆𝜓)
 
Theoremelqsn0 8356 A quotient set does not contain the empty set. (Contributed by NM, 24-Aug-1995.)
((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅)
 
Theoremecelqsdm 8357 Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)
((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → 𝐵𝐴)
 
Theoremxpider 8358 A Cartesian square is an equivalence relation (in general, it is not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝐴 × 𝐴) Er 𝐴
 
Theoremiiner 8359* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝑅 Er 𝐵) → 𝑥𝐴 𝑅 Er 𝐵)
 
Theoremriiner 8360* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
(∀𝑥𝐴 𝑅 Er 𝐵 → ((𝐵 × 𝐵) ∩ 𝑥𝐴 𝑅) Er 𝐵)
 
Theoremerinxp 8361 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 8362 Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)
(((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴)))
 
Theoremqsinxp 8363 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))))
 
Theoremqsdisj 8364 Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐵 ∈ (𝐴 / 𝑅))    &   (𝜑𝐶 ∈ (𝐴 / 𝑅))       (𝜑 → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))
 
Theoremqsdisj2 8365* A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.)
(𝑅 Er 𝑋Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥)
 
Theoremqsel 8366 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 𝑋𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)
 
Theoremuniinqs 8367 Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4852. (Contributed by FL, 25-May-2007.) (Proof shortened by Mario Carneiro, 11-Jul-2014.)
𝑅 Er 𝑋       ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝐵𝐶) = ( 𝐵 𝐶))
 
Theoremqliftlem 8368* 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)       ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
 
Theoremqliftrel 8369* 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)       (𝜑𝐹 ⊆ ((𝑋 / 𝑅) × 𝑌))
 
Theoremqliftel 8370* Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)       (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶𝑅𝑥𝐷 = 𝐴)))
 
Theoremqliftel1 8371* Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)       ((𝜑𝑥𝑋) → [𝑥]𝑅𝐹𝐴)
 
Theoremqliftfun 8372* 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 (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)    &   (𝑥 = 𝑦𝐴 = 𝐵)       (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵)))
 
Theoremqliftfund 8373* 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 (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)    &   (𝑥 = 𝑦𝐴 = 𝐵)    &   ((𝜑𝑥𝑅𝑦) → 𝐴 = 𝐵)       (𝜑 → Fun 𝐹)
 
Theoremqliftfuns 8374* 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 (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)       (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑧(𝑦𝑅𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)))
 
Theoremqliftf 8375* The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)       (𝜑 → (Fun 𝐹𝐹:(𝑋 / 𝑅)⟶𝑌))
 
Theoremqliftval 8376* The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)    &   (𝑥 = 𝐶𝐴 = 𝐵)    &   (𝜑 → Fun 𝐹)       ((𝜑𝐶𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵)
 
Theoremecoptocl 8377* 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))    &   ((𝑥𝐵𝑦𝐶) → 𝜑)       (𝐴𝑆𝜓)
 
Theorem2ecoptocl 8378* 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))    &   (((𝑥𝐶𝑦𝐷) ∧ (𝑧𝐶𝑤𝐷)) → 𝜑)       ((𝐴𝑆𝐵𝑆) → 𝜒)
 
Theorem3ecoptocl 8379* 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))    &   (((𝑥𝐷𝑦𝐷) ∧ (𝑧𝐷𝑤𝐷) ∧ (𝑣𝐷𝑢𝐷)) → 𝜑)       ((𝐴𝑆𝐵𝑆𝐶𝑆) → 𝜃)
 
Theorembrecop 8380* 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⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] 𝑦 = [⟨𝑣, 𝑢⟩] ) ∧ 𝜑))}    &   ((((𝑧𝐺𝑤𝐺) ∧ (𝐴𝐺𝐵𝐺)) ∧ ((𝑣𝐺𝑢𝐺) ∧ (𝐶𝐺𝐷𝐺))) → (([⟨𝑧, 𝑤⟩] = [⟨𝐴, 𝐵⟩] ∧ [⟨𝑣, 𝑢⟩] = [⟨𝐶, 𝐷⟩] ) → (𝜑𝜓)))       (((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)) → ([⟨𝐴, 𝐵⟩] [⟨𝐶, 𝐷⟩] 𝜓))
 
Theorembrecop2 8381 Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.) (Revised by AV, 12-Jul-2022.)
dom = (𝐺 × 𝐺)    &   𝐻 = ((𝐺 × 𝐺) / )    &   𝑅 ⊆ (𝐻 × 𝐻)    &    ⊆ (𝐺 × 𝐺)    &    ¬ ∅ ∈ 𝐺    &   dom + = (𝐺 × 𝐺)    &   (((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)) → ([⟨𝐴, 𝐵⟩] 𝑅[⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) (𝐵 + 𝐶)))       ([⟨𝐴, 𝐵⟩] 𝑅[⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) (𝐵 + 𝐶))
 
Theoremeroveu 8382* Lemma for erov 8384 and eroprf 8385. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝐽 = (𝐴 / 𝑅)    &   𝐾 = (𝐵 / 𝑆)    &   (𝜑𝑇𝑍)    &   (𝜑𝑅 Er 𝑈)    &   (𝜑𝑆 Er 𝑉)    &   (𝜑𝑇 Er 𝑊)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑+ :(𝐴 × 𝐵)⟶𝐶)    &   ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))       ((𝜑 ∧ (𝑋𝐽𝑌𝐾)) → ∃!𝑧𝑝𝐴𝑞𝐵 ((𝑋 = [𝑝]𝑅𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
 
Theoremerovlem 8383* Lemma for erov 8384 and eroprf 8385. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
𝐽 = (𝐴 / 𝑅)    &   𝐾 = (𝐵 / 𝑆)    &   (𝜑𝑇𝑍)    &   (𝜑𝑅 Er 𝑈)    &   (𝜑𝑆 Er 𝑉)    &   (𝜑𝑇 Er 𝑊)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑+ :(𝐴 × 𝐵)⟶𝐶)    &   ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))    &    = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}       (𝜑 = (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
 
Theoremerov 8384* The value 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)    &   (𝜑𝑇𝑍)    &   (𝜑𝑅 Er 𝑈)    &   (𝜑𝑆 Er 𝑉)    &   (𝜑𝑇 Er 𝑊)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑+ :(𝐴 × 𝐵)⟶𝐶)    &   ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))    &    = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}    &   (𝜑𝑅𝑋)    &   (𝜑𝑆𝑌)       ((𝜑𝑃𝐴𝑄𝐵) → ([𝑃]𝑅 [𝑄]𝑆) = [(𝑃 + 𝑄)]𝑇)
 
Theoremeroprf 8385* 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)    &   (𝜑𝑇𝑍)    &   (𝜑𝑅 Er 𝑈)    &   (𝜑𝑆 Er 𝑉)    &   (𝜑𝑇 Er 𝑊)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑+ :(𝐴 × 𝐵)⟶𝐶)    &   ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))    &    = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}    &   (𝜑𝑅𝑋)    &   (𝜑𝑆𝑌)    &   𝐿 = (𝐶 / 𝑇)       (𝜑 :(𝐽 × 𝐾)⟶𝐿)
 
Theoremerov2 8386* The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐽 = (𝐴 / )    &    = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐴 ((𝑥 = [𝑝] 𝑦 = [𝑞] ) ∧ 𝑧 = [(𝑝 + 𝑞)] )}    &   (𝜑𝑋)    &   (𝜑 Er 𝑈)    &   (𝜑𝐴𝑈)    &   (𝜑+ :(𝐴 × 𝐴)⟶𝐴)    &   ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐴𝑢𝐴))) → ((𝑟 𝑠𝑡 𝑢) → (𝑟 + 𝑡) (𝑠 + 𝑢)))       ((𝜑𝑃𝐴𝑄𝐴) → ([𝑃] [𝑄] ) = [(𝑃 + 𝑄)] )
 
Theoremeroprf2 8387* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐽 = (𝐴 / )    &    = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐴 ((𝑥 = [𝑝] 𝑦 = [𝑞] ) ∧ 𝑧 = [(𝑝 + 𝑞)] )}    &   (𝜑𝑋)    &   (𝜑 Er 𝑈)    &   (𝜑𝐴𝑈)    &   (𝜑+ :(𝐴 × 𝐴)⟶𝐴)    &   ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐴𝑢𝐴))) → ((𝑟 𝑠𝑡 𝑢) → (𝑟 + 𝑡) (𝑠 + 𝑢)))       (𝜑 :(𝐽 × 𝐽)⟶𝐽)
 
Theoremecopoveq 8388* 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)))}       (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝐶, 𝐷⟩ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
 
Theoremecopovsym 8389* Assuming the operation 𝐹 is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   (𝑥 + 𝑦) = (𝑦 + 𝑥)       (𝐴 𝐵𝐵 𝐴)
 
Theoremecopovtrn 8390* Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is transitive. (Contributed by NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   (𝑥 + 𝑦) = (𝑦 + 𝑥)    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))    &   ((𝑥𝑆𝑦𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))       ((𝐴 𝐵𝐵 𝐶) → 𝐴 𝐶)
 
Theoremecopover 8391* Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (Proof shortened by AV, 1-May-2021.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   (𝑥 + 𝑦) = (𝑦 + 𝑥)    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))    &   ((𝑥𝑆𝑦𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))        Er (𝑆 × 𝑆)
 
Theoremeceqoveq 8392* Equality of equivalence relation in terms of an operation. (Contributed by NM, 15-Feb-1996.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
Er (𝑆 × 𝑆)    &   dom + = (𝑆 × 𝑆)    &    ¬ ∅ ∈ 𝑆    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝐶, 𝐷⟩ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))       ((𝐴𝑆𝐶𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
 
Theoremecovcom 8393* Lemma used to transfer a commutative law via an equivalence relation. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
𝐶 = ((𝑆 × 𝑆) / )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐷, 𝐺⟩] )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑥𝑆𝑦𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = [⟨𝐻, 𝐽⟩] )    &   𝐷 = 𝐻    &   𝐺 = 𝐽       ((𝐴𝐶𝐵𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremecovass 8394* Lemma used to transfer an associative law via an equivalence relation. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
𝐷 = ((𝑆 × 𝑆) / )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐺, 𝐻⟩] )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑁, 𝑄⟩] )    &   (((𝐺𝑆𝐻𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑁𝑆𝑄𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝐺𝑆𝐻𝑆))    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑁𝑆𝑄𝑆))    &   𝐽 = 𝐿    &   𝐾 = 𝑀       ((𝐴𝐷𝐵𝐷𝐶𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
 
Theoremecovdi 8395* Lemma used to transfer a distributive law via an equivalence relation. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)
𝐷 = ((𝑆 × 𝑆) / )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑀, 𝑁⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑀𝑆𝑁𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) = [⟨𝑊, 𝑋⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] ) = [⟨𝑌, 𝑍⟩] )    &   (((𝑊𝑆𝑋𝑆) ∧ (𝑌𝑆𝑍𝑆)) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑀𝑆𝑁𝑆))    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝑊𝑆𝑋𝑆))    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑌𝑆𝑍𝑆))    &   𝐻 = 𝐾    &   𝐽 = 𝐿       ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 
2.4.22  The mapping operation
 
Syntaxcmap 8396 Extend the definition of a class to include the mapping operation. (Read for 𝐴m 𝐵, "the set of all functions that map from 𝐵 to 𝐴.)
class m
 
Syntaxcpm 8397 Extend the definition of a class to include the partial mapping operation. (Read for 𝐴pm 𝐵, "the set of all partial functions that map from 𝐵 to 𝐴.)
class pm
 
Definitiondf-map 8398* Define the mapping operation or set exponentiation. The set of all functions that map from 𝐵 to 𝐴 is written (𝐴m 𝐵) (see mapval 8408). Many authors write 𝐴 followed by 𝐵 as a superscript for this operation and rely on context to avoid confusion other exponentiation operations (e.g., Definition 10.42 of [TakeutiZaring] p. 95). Other authors show 𝐵 as a prefixed superscript, which is read "𝐴 pre 𝐵 " (e.g., definition of [Enderton] p. 52). Definition 8.21 of [Eisenberg] p. 125 uses the notation Map(𝐵, 𝐴) for our (𝐴m 𝐵). The up-arrow is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976). We adopt the first case of his notation (simple exponentiation) and subscript it with m to distinguish it from other kinds of exponentiation. (Contributed by NM, 8-Dec-2003.)
m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑦𝑥})
 
Definitiondf-pm 8399* Define the partial mapping operation. A partial function from 𝐵 to 𝐴 is a function from a subset of 𝐵 to 𝐴. The set of all partial functions from 𝐵 to 𝐴 is written (𝐴pm 𝐵) (see pmvalg 8407). A notation for this operation apparently does not appear in the literature. We use pm to distinguish it from the less general set exponentiation operation m (df-map 8398). See mapsspm 8430 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)
pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
 
Theoremmapprc 8400* When 𝐴 is a proper class, the class of all functions mapping 𝐴 to 𝐵 is empty. Exercise 4.41 of [Mendelson] p. 255. (Contributed by NM, 8-Dec-2003.)
𝐴 ∈ V → {𝑓𝑓:𝐴𝐵} = ∅)
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