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Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2ecoptocl 6601* 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 6602* 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 6603* 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 6604* Lemma for eroprf 6606. (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 6605* Lemma for eroprf 6606. (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 6606* 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 6607* 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 6608* 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 6609* 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 6610* 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 6611* 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 6612* 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 6613* 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 6614* 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 6615* 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 6616* 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 6617* 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 6618* 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 6619* 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 6620* Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6621 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 6621* 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 6622* Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6623 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 6623* 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 6624* Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6625 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 6625* 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 6626 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 6627 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 6628* Define the mapping operation or set exponentiation. The set of all functions that map from 𝐵 to 𝐴 is written (𝐴𝑚 𝐵) (see mapval 6638). 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 6629* 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 6637). 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 6628) . See mapsspm 6660 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)
pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
 
Theoremmapprc 6630* 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 6631* 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 6632* 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 6633 Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
𝑚 Fn (V × V)
 
Theoremfnpm 6634 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
pm Fn (V × V)
 
Theoremreldmmap 6635 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Rel dom ↑𝑚
 
Theoremmapvalg 6636* 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 6637* 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 6638* 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 6639 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 6640 Deduction form of elmapg 6639. (Contributed by BJ, 11-Apr-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐶 ∈ (𝐴𝑚 𝐵) ↔ 𝐶:𝐵𝐴))
 
Theoremmapdm0 6641 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 6642 The predicate "is a partial function". (Contributed by Mario Carneiro, 14-Nov-2013.)
((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴pm 𝐵) ↔ (Fun 𝐶𝐶 ⊆ (𝐵 × 𝐴))))
 
Theoremelpm2g 6643 The predicate "is a partial function". (Contributed by NM, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊) → (𝐹 ∈ (𝐴pm 𝐵) ↔ (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵)))
 
Theoremelpm2r 6644 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐹:𝐶𝐴𝐶𝐵)) → 𝐹 ∈ (𝐴pm 𝐵))
 
Theoremelpmi 6645 A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)
(𝐹 ∈ (𝐴pm 𝐵) → (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵))
 
Theorempmfun 6646 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 6647 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 6648 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 6649 A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
(𝐴 ∈ (𝐵𝑚 𝐶) → 𝐴 Fn 𝐶)
 
Theoremelmapfun 6650 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 6651 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 6652 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊𝐹:𝐴𝐵) → 𝐹 ∈ (𝐵pm 𝐴))
 
Theorempmss12g 6653 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
(((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝐴pm 𝐵) ⊆ (𝐶pm 𝐷))
 
Theorempmresg 6654 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 6655 Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹 ∈ (𝐴𝑚 𝐵) ↔ 𝐹:𝐵𝐴)
 
Theoremmapval2 6656* 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 6657 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 6658 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 6659 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 6660 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 6661 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 6662 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 6663* Special case of fvmpt 5573 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 6664 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 6665 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 6666 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 6667 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 6668* 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 6669 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 6670* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝐵𝑉𝐼𝑊) → 𝐹:𝐵⟶(𝐵𝑚 𝐼))
 
Theoremfvdiagfn 6671* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝐼𝑊𝑋𝐵) → (𝐹𝑋) = (𝐼 × {𝑋}))
 
Theoremmapsnconst 6672 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 6673* 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 6674* 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 6675* Explicit bijection in the reverse of mapsnf1o2 6674. (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 6676 Extend class notation to include infinite Cartesian products.
class X𝑥𝐴 𝐵
 
Definitiondf-ixp 6677* 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 6678* Eliminate the expression {𝑥𝑥𝐴} in df-ixp 6677, 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 6679* 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 6680* Membership in an infinite Cartesian product. See df-ixp 6677 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)
(𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremfvixp 6681* Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝑥 = 𝐶𝐵 = 𝐷)       ((𝐹X𝑥𝐴 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
 
Theoremixpfn 6682* A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)
(𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
 
Theoremelixp 6683* Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
𝐹 ∈ V       (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremelixpconst 6684* Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by NM, 12-Apr-2008.)
𝐹 ∈ V       (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
 
Theoremixpconstg 6685* Infinite Cartesian product of a constant 𝐵. (Contributed by Mario Carneiro, 11-Jan-2015.)
((𝐴𝑉𝐵𝑊) → X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴))
 
Theoremixpconst 6686* Infinite Cartesian product of a constant 𝐵. (Contributed by NM, 28-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴)
 
Theoremixpeq1 6687* Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
(𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
 
Theoremixpeq1d 6688* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐴 = 𝐵)       (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
 
Theoremss2ixp 6689 Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)
(∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
 
Theoremixpeq2 6690 Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
(∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremixpeq2dva 6691* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremixpeq2dv 6692* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremcbvixp 6693* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
 
Theoremcbvixpv 6694* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑥 = 𝑦𝐵 = 𝐶)       X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
 
Theoremnfixpxy 6695* Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon, 15-Feb-2023.)
𝑦𝐴    &   𝑦𝐵       𝑦X𝑥𝐴 𝐵
 
Theoremnfixp1 6696 The index variable in an indexed Cartesian product is not free. (Contributed by Jeff Madsen, 19-Jun-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥X𝑥𝐴 𝐵
 
Theoremixpprc 6697* A cartesian product of proper-class many sets is empty, because any function in the cartesian product has to be a set with domain 𝐴, which is not possible for a proper class domain. (Contributed by Mario Carneiro, 25-Jan-2015.)
𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
 
Theoremixpf 6698* A member of an infinite Cartesian product maps to the indexed union of the product argument. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)
(𝐹X𝑥𝐴 𝐵𝐹:𝐴 𝑥𝐴 𝐵)
 
Theoremuniixp 6699* The union of an infinite Cartesian product is included in a Cartesian product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 24-Jun-2015.)
X𝑥𝐴 𝐵 ⊆ (𝐴 × 𝑥𝐴 𝐵)
 
Theoremixpexgg 6700* The existence of an infinite Cartesian product. 𝑥 is normally a free-variable parameter in 𝐵. Remark in Enderton p. 54. (Contributed by NM, 28-Sep-2006.) (Revised by Jim Kingdon, 15-Feb-2023.)
((𝐴𝑊 ∧ ∀𝑥𝐴 𝐵𝑉) → X𝑥𝐴 𝐵 ∈ V)
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