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Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiinerm 6601* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝑅 Er 𝐵) → 𝑥𝐴 𝑅 Er 𝐵)
 
Theoremriinerm 6602* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝑅 Er 𝐵) → ((𝐵 × 𝐵) ∩ 𝑥𝐴 𝑅) Er 𝐵)
 
Theoremerinxp 6603 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 6604 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 6605 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 6606 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)
 
Theoremqliftlem 6607* 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 6608* 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 6609* 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 6610* 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 6611* 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 6612* 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 6613* 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 6614* The domain and codomain 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 6615* 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 6616* 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 6617* 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 6618* 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 6619* 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))
 
Theoremeroveu 6620* Lemma for eroprf 6622. (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 6621* Lemma for eroprf 6622. (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))))
 
Theoremeroprf 6622* 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)
 
Theoremeroprf2 6623* 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 6624* 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 6625* 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 6626* 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 6627* 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.)
= {⟨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)
 
Theoremecopovsymg 6628* Assuming the operation 𝐹 is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))       (𝐴 𝐵𝐵 𝐴)
 
Theoremecopovtrng 6629* 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 Jim Kingdon, 1-Sep-2019.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))       ((𝐴 𝐵𝐵 𝐶) → 𝐴 𝐶)
 
Theoremecopoverg 6630* 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 Jim Kingdon, 1-Sep-2019.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))        Er (𝑆 × 𝑆)
 
Theoremth3qlem1 6631* Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption. (Contributed by NM, 3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Er 𝑆    &   (((𝑦𝑆𝑤𝑆) ∧ (𝑧𝑆𝑣𝑆)) → ((𝑦 𝑤𝑧 𝑣) → (𝑦 + 𝑧) (𝑤 + 𝑣)))       ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → ∃*𝑥𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ))
 
Theoremth3qlem2 6632* Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
∈ V    &    Er (𝑆 × 𝑆)    &   ((((𝑤𝑆𝑣𝑆) ∧ (𝑢𝑆𝑡𝑆)) ∧ ((𝑠𝑆𝑓𝑆) ∧ (𝑔𝑆𝑆))) → ((⟨𝑤, 𝑣𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓𝑔, ⟩) → (⟨𝑤, 𝑣+𝑠, 𝑓⟩) (⟨𝑢, 𝑡+𝑔, ⟩)))       ((𝐴 ∈ ((𝑆 × 𝑆) / ) ∧ 𝐵 ∈ ((𝑆 × 𝑆) / )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] 𝐵 = [⟨𝑢, 𝑡⟩] ) ∧ 𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))
 
Theoremth3qcor 6633* Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)
∈ V    &    Er (𝑆 × 𝑆)    &   ((((𝑤𝑆𝑣𝑆) ∧ (𝑢𝑆𝑡𝑆)) ∧ ((𝑠𝑆𝑓𝑆) ∧ (𝑔𝑆𝑆))) → ((⟨𝑤, 𝑣𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓𝑔, ⟩) → (⟨𝑤, 𝑣+𝑠, 𝑓⟩) (⟨𝑢, 𝑡+𝑔, ⟩)))    &   𝐺 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ∧ 𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))}       Fun 𝐺
 
Theoremth3q 6634* Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
∈ V    &    Er (𝑆 × 𝑆)    &   ((((𝑤𝑆𝑣𝑆) ∧ (𝑢𝑆𝑡𝑆)) ∧ ((𝑠𝑆𝑓𝑆) ∧ (𝑔𝑆𝑆))) → ((⟨𝑤, 𝑣𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓𝑔, ⟩) → (⟨𝑤, 𝑣+𝑠, 𝑓⟩) (⟨𝑢, 𝑡+𝑔, ⟩)))    &   𝐺 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ∧ 𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))}       (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] )
 
Theoremoviec 6635* Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See iset.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.)
(((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → 𝐻 ∈ (𝑆 × 𝑆))    &   (((𝑎𝑆𝑏𝑆) ∧ (𝑔𝑆𝑆)) → 𝐾 ∈ (𝑆 × 𝑆))    &   (((𝑐𝑆𝑑𝑆) ∧ (𝑡𝑆𝑠𝑆)) → 𝐿 ∈ (𝑆 × 𝑆))    &    ∈ V    &    Er (𝑆 × 𝑆)    &    = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}    &   (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → (𝜑𝜓))    &   (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → (𝜑𝜒))    &    + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝐽))}    &   (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝐽 = 𝐾)    &   (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝐽 = 𝐿)    &   (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝐽 = 𝐻)    &    = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑄𝑦𝑄) ∧ ∃𝑎𝑏𝑐𝑑((𝑥 = [⟨𝑎, 𝑏⟩] 𝑦 = [⟨𝑐, 𝑑⟩] ) ∧ 𝑧 = [(⟨𝑎, 𝑏+𝑐, 𝑑⟩)] ))}    &   𝑄 = ((𝑆 × 𝑆) / )    &   ((((𝑎𝑆𝑏𝑆) ∧ (𝑐𝑆𝑑𝑆)) ∧ ((𝑔𝑆𝑆) ∧ (𝑡𝑆𝑠𝑆))) → ((𝜓𝜒) → 𝐾 𝐿))       (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] [⟨𝐶, 𝐷⟩] ) = [𝐻] )
 
Theoremecovcom 6636* Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6637 instead. (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))
 
Theoremecovicom 6637* Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.)
𝐶 = ((𝑆 × 𝑆) / )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐷, 𝐺⟩] )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑥𝑆𝑦𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = [⟨𝐻, 𝐽⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → 𝐷 = 𝐻)    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → 𝐺 = 𝐽)       ((𝐴𝐶𝐵𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremecovass 6638* Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6639 will be more useful. (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)))
 
Theoremecoviass 6639* Lemma used to transfer an associative law via an equivalence relation. (Contributed by Jim Kingdon, 16-Sep-2019.)
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))    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐽 = 𝐿)    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐾 = 𝑀)       ((𝐴𝐷𝐵𝐷𝐶𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
 
Theoremecovdi 6640* Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6641 will be more helpful. (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)))
 
Theoremecovidi 6641* Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.)
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))    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐻 = 𝐾)    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐽 = 𝐿)       ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 
2.6.26  The mapping operation
 
Syntaxcmap 6642 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 6643 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 6644* Define the mapping operation or set exponentiation. The set of all functions that map from 𝐵 to 𝐴 is written (𝐴𝑚 𝐵) (see mapval 6654). 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 6645* 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 6653). 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 6644) . See mapsspm 6676 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)
pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
 
Theoremmapprc 6646* 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} = ∅)
 
Theorempmex 6647* The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)
((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (Fun 𝑓𝑓 ⊆ (𝐴 × 𝐵))} ∈ V)
 
Theoremmapex 6648* The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.)
((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐴𝐵} ∈ V)
 
Theoremfnmap 6649 Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
𝑚 Fn (V × V)
 
Theoremfnpm 6650 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
pm Fn (V × V)
 
Theoremreldmmap 6651 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Rel dom ↑𝑚
 
Theoremmapvalg 6652* The value of set exponentiation. (𝐴𝑚 𝐵) is the set of all functions that map from 𝐵 to 𝐴. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴𝐶𝐵𝐷) → (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴})
 
Theorempmvalg 6653* The value of the partial mapping operation. (𝐴pm 𝐵) is the set of all partial functions that map from 𝐵 to 𝐴. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴𝐶𝐵𝐷) → (𝐴pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
 
Theoremmapval 6654* The value of set exponentiation (inference version). (𝐴𝑚 𝐵) is the set of all functions that map from 𝐵 to 𝐴. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴}
 
Theoremelmapg 6655 Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴𝑚 𝐵) ↔ 𝐶:𝐵𝐴))
 
Theoremelmapd 6656 Deduction form of elmapg 6655. (Contributed by BJ, 11-Apr-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐶 ∈ (𝐴𝑚 𝐵) ↔ 𝐶:𝐵𝐴))
 
Theoremmapdm0 6657 The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux, 3-Dec-2021.)
(𝐵𝑉 → (𝐵𝑚 ∅) = {∅})
 
Theoremelpmg 6658 The predicate "is a partial function". (Contributed by Mario Carneiro, 14-Nov-2013.)
((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴pm 𝐵) ↔ (Fun 𝐶𝐶 ⊆ (𝐵 × 𝐴))))
 
Theoremelpm2g 6659 The predicate "is a partial function". (Contributed by NM, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊) → (𝐹 ∈ (𝐴pm 𝐵) ↔ (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵)))
 
Theoremelpm2r 6660 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐹:𝐶𝐴𝐶𝐵)) → 𝐹 ∈ (𝐴pm 𝐵))
 
Theoremelpmi 6661 A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)
(𝐹 ∈ (𝐴pm 𝐵) → (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵))
 
Theorempmfun 6662 A partial function is a function. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐹 ∈ (𝐴pm 𝐵) → Fun 𝐹)
 
Theoremelmapex 6663 Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.)
(𝐴 ∈ (𝐵𝑚 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
 
Theoremelmapi 6664 A mapping is a function, forward direction only with superfluous antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ (𝐵𝑚 𝐶) → 𝐴:𝐶𝐵)
 
Theoremelmapfn 6665 A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
(𝐴 ∈ (𝐵𝑚 𝐶) → 𝐴 Fn 𝐶)
 
Theoremelmapfun 6666 A mapping is always a function. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
(𝐴 ∈ (𝐵𝑚 𝐶) → Fun 𝐴)
 
Theoremelmapssres 6667 A restricted mapping is a mapping. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
((𝐴 ∈ (𝐵𝑚 𝐶) ∧ 𝐷𝐶) → (𝐴𝐷) ∈ (𝐵𝑚 𝐷))
 
Theoremfpmg 6668 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊𝐹:𝐴𝐵) → 𝐹 ∈ (𝐵pm 𝐴))
 
Theorempmss12g 6669 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
(((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝐴pm 𝐵) ⊆ (𝐶pm 𝐷))
 
Theorempmresg 6670 Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → (𝐹𝐵) ∈ (𝐴pm 𝐵))
 
Theoremelmap 6671 Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹 ∈ (𝐴𝑚 𝐵) ↔ 𝐹:𝐵𝐴)
 
Theoremmapval2 6672* Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝑚 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵})
 
Theoremelpm 6673 The predicate "is a partial function". (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹 ∈ (𝐴pm 𝐵) ↔ (Fun 𝐹𝐹 ⊆ (𝐵 × 𝐴)))
 
Theoremelpm2 6674 The predicate "is a partial function". (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹 ∈ (𝐴pm 𝐵) ↔ (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵))
 
Theoremfpm 6675 A total function is a partial function. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹:𝐴𝐵𝐹 ∈ (𝐵pm 𝐴))
 
Theoremmapsspm 6676 Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)
(𝐴𝑚 𝐵) ⊆ (𝐴pm 𝐵)
 
Theorempmsspw 6677 Partial maps are a subset of the power set of the Cartesian product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
 
Theoremmapsspw 6678 Set exponentiation is a subset of the power set of the Cartesian product of its arguments. (Contributed by NM, 8-Dec-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
 
Theoremfvmptmap 6679* Special case of fvmpt 5589 for operator theorems. (Contributed by NM, 27-Nov-2007.)
𝐶 ∈ V    &   𝐷 ∈ V    &   𝑅 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥 ∈ (𝑅𝑚 𝐷) ↦ 𝐵)       (𝐴:𝐷𝑅 → (𝐹𝐴) = 𝐶)
 
Theoremmap0e 6680 Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝐴𝑉 → (𝐴𝑚 ∅) = 1o)
 
Theoremmap0b 6681 Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴 ≠ ∅ → (∅ ↑𝑚 𝐴) = ∅)
 
Theoremmap0g 6682 Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
 
Theoremmap0 6683 Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐴𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅))
 
Theoremmapsn 6684* The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}
 
Theoremmapss 6685 Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐵𝑉𝐴𝐵) → (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
 
Theoremfdiagfn 6686* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝐵𝑉𝐼𝑊) → 𝐹:𝐵⟶(𝐵𝑚 𝐼))
 
Theoremfvdiagfn 6687* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝐼𝑊𝑋𝐵) → (𝐹𝑋) = (𝐼 × {𝑋}))
 
Theoremmapsnconst 6688 Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑆 = {𝑋}    &   𝐵 ∈ V    &   𝑋 ∈ V       (𝐹 ∈ (𝐵𝑚 𝑆) → 𝐹 = (𝑆 × {(𝐹𝑋)}))
 
Theoremmapsncnv 6689* Expression for the inverse of the canonical map between a set and its set of singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝑆 = {𝑋}    &   𝐵 ∈ V    &   𝑋 ∈ V    &   𝐹 = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))       𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
 
Theoremmapsnf1o2 6690* Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝑆 = {𝑋}    &   𝐵 ∈ V    &   𝑋 ∈ V    &   𝐹 = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))       𝐹:(𝐵𝑚 𝑆)–1-1-onto𝐵
 
Theoremmapsnf1o3 6691* Explicit bijection in the reverse of mapsnf1o2 6690. (Contributed by Stefan O'Rear, 24-Mar-2015.)
𝑆 = {𝑋}    &   𝐵 ∈ V    &   𝑋 ∈ V    &   𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))       𝐹:𝐵1-1-onto→(𝐵𝑚 𝑆)
 
2.6.27  Infinite Cartesian products
 
Syntaxcixp 6692 Extend class notation to include infinite Cartesian products.
class X𝑥𝐴 𝐵
 
Definitiondf-ixp 6693* Definition of infinite Cartesian product of [Enderton] p. 54. Enderton uses a bold "X" with 𝑥𝐴 written underneath or as a subscript, as does Stoll p. 47. Some books use a capital pi, but we will reserve that notation for products of numbers. Usually 𝐵 represents a class expression containing 𝑥 free and thus can be thought of as 𝐵(𝑥). Normally, 𝑥 is not free in 𝐴, although this is not a requirement of the definition. (Contributed by NM, 28-Sep-2006.)
X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
 
Theoremdfixp 6694* Eliminate the expression {𝑥𝑥𝐴} in df-ixp 6693, under the assumption that 𝐴 and 𝑥 are disjoint. This way, we can say that 𝑥 is bound in X𝑥𝐴𝐵 even if it appears free in 𝐴. (Contributed by Mario Carneiro, 12-Aug-2016.)
X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
 
Theoremixpsnval 6695* The value of an infinite Cartesian product with a singleton. (Contributed by AV, 3-Dec-2018.)
(𝑋𝑉X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓𝑋) ∈ 𝑋 / 𝑥𝐵)})
 
Theoremelixp2 6696* Membership in an infinite Cartesian product. See df-ixp 6693 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)
(𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremfvixp 6697* Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝑥 = 𝐶𝐵 = 𝐷)       ((𝐹X𝑥𝐴 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
 
Theoremixpfn 6698* A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)
(𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
 
Theoremelixp 6699* Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
𝐹 ∈ V       (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremelixpconst 6700* Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by NM, 12-Apr-2008.)
𝐹 ∈ V       (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
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