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Theorem List for Metamath Proof Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2ecoptocl 7601* 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 7602* 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 7603* 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 7604 Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.)
∈ V    &   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 7605* Lemma for erov 7607 and eroprf 7608. (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 7606* Lemma for erov 7607 and eroprf 7608. (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 7607* 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 7608* 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 7609* 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 7610* 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 7611* 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 7612* 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 7613* 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 7614* 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)
 
TheoremecopoverOLD 7615* Obsolete proof of ecopover 7614 as of 1-May-2021. 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 modification is discouraged.) (New usage is discouraged.)
= {⟨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 7616* 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 7617* 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 7618* 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 7619* 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)))
 
2.4.21  The mapping operation
 
Syntaxcmap 7620 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 7621 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 7622* Define the mapping operation or set exponentiation. The set of all functions that map from 𝐵 to 𝐴 is written (𝐴𝑚 𝐵) (see mapval 7632). 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 7623* 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 7631). 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 7622) . See mapsspm 7653 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)
pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
 
Theoremmapprc 7624* 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 7625* 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 7626* 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 7627 Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
𝑚 Fn (V × V)
 
Theoremfnpm 7628 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
pm Fn (V × V)
 
Theoremreldmmap 7629 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Rel dom ↑𝑚
 
Theoremmapvalg 7630* 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 7631* 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 7632* 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 7633 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 7634 Deduction form of elmapg 7633. (Contributed by BJ, 11-Apr-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐶 ∈ (𝐴𝑚 𝐵) ↔ 𝐶:𝐵𝐴))
 
Theoremelpmg 7635 The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)
((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴pm 𝐵) ↔ (Fun 𝐶𝐶 ⊆ (𝐵 × 𝐴))))
 
Theoremelpm2g 7636 The predicate "is a partial function." (Contributed by NM, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊) → (𝐹 ∈ (𝐴pm 𝐵) ↔ (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵)))
 
Theoremelpm2r 7637 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐹:𝐶𝐴𝐶𝐵)) → 𝐹 ∈ (𝐴pm 𝐵))
 
Theoremelpmi 7638 A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)
(𝐹 ∈ (𝐴pm 𝐵) → (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵))
 
Theorempmfun 7639 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 7640 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 7641 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 7642 A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
(𝐴 ∈ (𝐵𝑚 𝐶) → 𝐴 Fn 𝐶)
 
Theoremelmapfun 7643 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 7644 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 7645 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊𝐹:𝐴𝐵) → 𝐹 ∈ (𝐵pm 𝐴))
 
Theorempmss12g 7646 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
(((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝐴pm 𝐵) ⊆ (𝐶pm 𝐷))
 
Theorempmresg 7647 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 7648 Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹 ∈ (𝐴𝑚 𝐵) ↔ 𝐹:𝐵𝐴)
 
Theoremmapval2 7649* 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 7650 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 7651 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 7652 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 7653 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 7654 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 7655 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 7656* Special case of fvmpt 6075 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 7657 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 7658 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 7659 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 7660 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 7661* 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 7662 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 7663* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝐵𝑉𝐼𝑊) → 𝐹:𝐵⟶(𝐵𝑚 𝐼))
 
Theoremfvdiagfn 7664* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝐼𝑊𝑋𝐵) → (𝐹𝑋) = (𝐼 × {𝑋}))
 
Theoremmapsnconst 7665 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 7666* 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 7667* 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 7668* Explicit bijection in the reverse of mapsnf1o2 7667. (Contributed by Stefan O'Rear, 24-Mar-2015.)
𝑆 = {𝑋}    &   𝐵 ∈ V    &   𝑋 ∈ V    &   𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))       𝐹:𝐵1-1-onto→(𝐵𝑚 𝑆)
 
Theoremralxpmap 7669* Quantification over functions in terms of quantification over values and punctured functions. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
(𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝜑𝜓))       (𝐽𝑇 → (∀𝑓 ∈ (𝑆𝑚 𝑇)𝜑 ↔ ∀𝑦𝑆𝑔 ∈ (𝑆𝑚 (𝑇 ∖ {𝐽}))𝜓))
 
2.4.22  Infinite Cartesian products
 
Syntaxcixp 7670 Extend class notation to include infinite Cartesian products.
class X𝑥𝐴 𝐵
 
Definitiondf-ixp 7671* 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 7672* Eliminate the expression {𝑥𝑥𝐴} in df-ixp 7671, 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 7673* 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 7674* Membership in an infinite Cartesian product. See df-ixp 7671 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)
(𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremfvixp 7675* Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝑥 = 𝐶𝐵 = 𝐷)       ((𝐹X𝑥𝐴 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
 
Theoremixpfn 7676* A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)
(𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
 
Theoremelixp 7677* Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
𝐹 ∈ V       (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremelixpconst 7678* Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by NM, 12-Apr-2008.)
𝐹 ∈ V       (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
 
Theoremixpconstg 7679* Infinite Cartesian product of a constant 𝐵. (Contributed by Mario Carneiro, 11-Jan-2015.)
((𝐴𝑉𝐵𝑊) → X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴))
 
Theoremixpconst 7680* Infinite Cartesian product of a constant 𝐵. (Contributed by NM, 28-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴)
 
Theoremixpeq1 7681* Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
(𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
 
Theoremixpeq1d 7682* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐴 = 𝐵)       (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
 
Theoremss2ixp 7683 Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)
(∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
 
Theoremixpeq2 7684 Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
(∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremixpeq2dva 7685* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremixpeq2dv 7686* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremcbvixp 7687* 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 7688* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑥 = 𝑦𝐵 = 𝐶)       X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
 
Theoremnfixp 7689 Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.)
𝑦𝐴    &   𝑦𝐵       𝑦X𝑥𝐴 𝐵
 
Theoremnfixp1 7690 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 7691* 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 7692* 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 7693* 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)
 
Theoremixpexg 7694* 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 Mario Carneiro, 25-Jan-2015.)
(∀𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ∈ V)
 
Theoremixpin 7695* The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
X𝑥𝐴 (𝐵𝐶) = (X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
 
Theoremixpiin 7696* The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.)
(𝐵 ≠ ∅ → X𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 X𝑥𝐴 𝐶)
 
Theoremixpint 7697* The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
(𝐵 ≠ ∅ → X𝑥𝐴 𝐵 = 𝑦𝐵 X𝑥𝐴 𝑦)
 
Theoremixp0x 7698 An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
X𝑥 ∈ ∅ 𝐴 = {∅}
 
Theoremixpssmap2g 7699* An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 7700 avoids ax-rep 4597. (Contributed by Mario Carneiro, 16-Nov-2014.)
( 𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
 
Theoremixpssmapg 7700* An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.)
(∀𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
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