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Theorem List for Metamath Proof Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-map 8701* Define the mapping operation or set exponentiation. The set of all functions that map from 𝐵 to 𝐴 is written (𝐴m 𝐵) (see mapval 8711). 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 8702* 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 8710). 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 8701). See mapsspm 8748 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)
pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
 
Theoremmapprc 8703* 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 8704* 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 8705* 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 8706 Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
m Fn (V × V)
 
Theoremfnpm 8707 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
pm Fn (V × V)
 
Theoremreldmmap 8708 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Rel dom ↑m
 
Theoremmapvalg 8709* 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 8710* 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 8711* 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 8712 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 8713 Deduction form of elmapg 8712. (Contributed by BJ, 11-Apr-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐶 ∈ (𝐴m 𝐵) ↔ 𝐶:𝐵𝐴))
 
Theoremmapdm0 8714 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 8715 The predicate "is a partial function". (Contributed by Mario Carneiro, 14-Nov-2013.)
((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴pm 𝐵) ↔ (Fun 𝐶𝐶 ⊆ (𝐵 × 𝐴))))
 
Theoremelpm2g 8716 The predicate "is a partial function". (Contributed by NM, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊) → (𝐹 ∈ (𝐴pm 𝐵) ↔ (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵)))
 
Theoremelpm2r 8717 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐹:𝐶𝐴𝐶𝐵)) → 𝐹 ∈ (𝐴pm 𝐵))
 
Theoremelpmi 8718 A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)
(𝐹 ∈ (𝐴pm 𝐵) → (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵))
 
Theorempmfun 8719 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 8720 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 8721 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)
 
Theoremmapfset 8722* If 𝐵 is a set, the value of the set exponentiation (𝐵m 𝐴) is the class of all functions from 𝐴 to 𝐵. Generalisation of mapvalg 8709 (which does not require ax-rep 5241) to arbitrary domains. Note that the class {𝑓𝑓:𝐴𝐵} can only contain set-functions, as opposed to arbitrary class-functions. When 𝐴 is a proper class, there can be no set-functions on it, so the above class is empty (see also fsetdmprc0 8727), hence a set. In this case, both sides of the equality in this theorem are the empty set. (Contributed by AV, 8-Aug-2024.)
(𝐵𝑉 → {𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴))
 
Theoremmapssfset 8723* The value of the set exponentiation (𝐵m 𝐴) is a subset of the class of functions from 𝐴 to 𝐵. (Contributed by AV, 10-Aug-2024.)
(𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵}
 
Theoremmapfoss 8724* The value of the set exponentiation (𝐵m 𝐴) is a superset of the set of all functions from 𝐴 onto 𝐵. (Contributed by AV, 7-Aug-2024.)
{𝑓𝑓:𝐴onto𝐵} ⊆ (𝐵m 𝐴)
 
Theoremfsetsspwxp 8725* The class of all functions from 𝐴 into 𝐵 is a subclass of the power class of the cartesion product of 𝐴 and 𝐵. (Contributed by AV, 13-Sep-2024.)
{𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
 
Theoremfset0 8726 The set of functions from the empty set is the singleton containing the empty set. (Contributed by AV, 13-Sep-2024.)
{𝑓𝑓:∅⟶𝐵} = {∅}
 
Theoremfsetdmprc0 8727* The set of functions with a proper class as domain is empty. (Contributed by AV, 22-Aug-2024.)
(𝐴 ∉ V → {𝑓𝑓 Fn 𝐴} = ∅)
 
Theoremfsetex 8728* The set of functions between two classes exists if the codomain exists. Generalization of mapex 8705 to arbitrary domains. (Contributed by AV, 14-Aug-2024.)
(𝐵𝑉 → {𝑓𝑓:𝐴𝐵} ∈ V)
 
Theoremf1setex 8729* The set of injections between two classes exists if the codomain exists. (Contributed by AV, 14-Aug-2024.)
(𝐵𝑉 → {𝑓𝑓:𝐴1-1𝐵} ∈ V)
 
Theoremfosetex 8730* The set of surjections between two classes exists (without any precondition). (Contributed by AV, 8-Aug-2024.)
{𝑓𝑓:𝐴onto𝐵} ∈ V
 
Theoremf1osetex 8731* The set of bijections between two classes exists. (Contributed by AV, 30-Mar-2024.) (Revised by AV, 8-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
{𝑓𝑓:𝐴1-1-onto𝐵} ∈ V
 
Theoremfsetfcdm 8732* The class of functions with a given domain and a given codomain is mapped, through evaluation at a point of the domain, into the codomain. (Contributed by AV, 15-Sep-2024.)
𝐹 = {𝑓𝑓:𝐴𝐵}    &   𝑆 = (𝑔𝐹 ↦ (𝑔𝑋))       (𝑋𝐴𝑆:𝐹𝐵)
 
Theoremfsetfocdm 8733* The class of functions with a given domain that is a set and a given codomain is mapped, through evaluation at a point of the domain, onto the codomain. (Contributed by AV, 15-Sep-2024.)
𝐹 = {𝑓𝑓:𝐴𝐵}    &   𝑆 = (𝑔𝐹 ↦ (𝑔𝑋))       ((𝐴𝑉𝑋𝐴) → 𝑆:𝐹onto𝐵)
 
Theoremfsetprcnex 8734* The class of all functions from a nonempty set 𝐴 into a proper class 𝐵 is not a set. If one of the preconditions is not fufilled, then {𝑓𝑓:𝐴𝐵} is a set, see fsetdmprc0 8727 for 𝐴 ∉ V, fset0 8726 for 𝐴 = ∅, and fsetex 8728 for 𝐵 ∈ V, see also fsetexb 8736. (Contributed by AV, 14-Sep-2024.) (Proof shortened by BJ, 15-Sep-2024.)
(((𝐴𝑉𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓𝑓:𝐴𝐵} ∉ V)
 
Theoremfsetcdmex 8735* The class of all functions from a nonempty set 𝐴 into a class 𝐵 is a set iff 𝐵 is a set . (Contributed by AV, 15-Sep-2024.)
((𝐴𝑉𝐴 ≠ ∅) → (𝐵 ∈ V ↔ {𝑓𝑓:𝐴𝐵} ∈ V))
 
Theoremfsetexb 8736* The class of all functions from a class 𝐴 into a class 𝐵 is a set iff 𝐵 is a set or 𝐴 is not a set or 𝐴 is empty. (Contributed by AV, 15-Sep-2024.)
({𝑓𝑓:𝐴𝐵} ∈ V ↔ (𝐴 ∉ V ∨ 𝐴 = ∅ ∨ 𝐵 ∈ V))
 
Theoremelmapfn 8737 A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
(𝐴 ∈ (𝐵m 𝐶) → 𝐴 Fn 𝐶)
 
Theoremelmapfun 8738 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 8739 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 8740 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊𝐹:𝐴𝐵) → 𝐹 ∈ (𝐵pm 𝐴))
 
Theorempmss12g 8741 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
(((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝐴pm 𝐵) ⊆ (𝐶pm 𝐷))
 
Theorempmresg 8742 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 8743 Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹 ∈ (𝐴m 𝐵) ↔ 𝐹:𝐵𝐴)
 
Theoremmapval2 8744* 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 8745 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 8746 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 8747 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 8748 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 8749 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 8750 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)
 
Theoremmapfvd 8751 The value of a function that maps from 𝐵 to 𝐴. (Contributed by AV, 2-Feb-2023.)
𝑀 = (𝐴m 𝐵)    &   (𝜑𝐹𝑀)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐹𝑋) ∈ 𝐴)
 
Theoremelmapresaun 8752 fresaun 6709 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) ∈ (𝐶m (𝐴𝐵)))
 
Theoremfvmptmap 8753* Special case of fvmpt 6944 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 8754 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.) (Proof shortened by AV, 14-Jul-2022.)
(𝐴𝑉 → (𝐴m ∅) = 1o)
 
Theoremmap0b 8755 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 8756 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 ≠ ∅)))
 
Theorem0map0sn0 8757 The set of mappings of the empty set to the empty set is the singleton containing the empty set. (Contributed by AV, 31-Mar-2024.)
(∅ ↑m ∅) = {∅}
 
Theoremmapsnd 8758* The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴m {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})
 
Theoremmap0 8759 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 8760* The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Proof shortened by AV, 17-Jul-2022.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴m {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}
 
Theoremmapss 8761 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 8762* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝐵𝑉𝐼𝑊) → 𝐹:𝐵⟶(𝐵m 𝐼))
 
Theoremfvdiagfn 8763* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝐼𝑊𝑋𝐵) → (𝐹𝑋) = (𝐼 × {𝑋}))
 
Theoremmapsnconst 8764 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 8765* 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 8766* 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 8767* Explicit bijection in the reverse of mapsnf1o2 8766. (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 8768* 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))
 
2.4.25  Infinite Cartesian products
 
Syntaxcixp 8769 Extend class notation to include infinite Cartesian products.
class X𝑥𝐴 𝐵
 
Definitiondf-ixp 8770* 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 8771* Eliminate the expression {𝑥𝑥𝐴} in df-ixp 8770, 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 8772* 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 8773* Membership in an infinite Cartesian product. See df-ixp 8770 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)
(𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremfvixp 8774* Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝑥 = 𝐶𝐵 = 𝐷)       ((𝐹X𝑥𝐴 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
 
Theoremixpfn 8775* A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)
(𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
 
Theoremelixp 8776* Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
𝐹 ∈ V       (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremelixpconst 8777* Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by NM, 12-Apr-2008.)
𝐹 ∈ V       (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
 
Theoremixpconstg 8778* Infinite Cartesian product of a constant 𝐵. (Contributed by Mario Carneiro, 11-Jan-2015.)
((𝐴𝑉𝐵𝑊) → X𝑥𝐴 𝐵 = (𝐵m 𝐴))
 
Theoremixpconst 8779* Infinite Cartesian product of a constant 𝐵. (Contributed by NM, 28-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       X𝑥𝐴 𝐵 = (𝐵m 𝐴)
 
Theoremixpeq1 8780* Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
(𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
 
Theoremixpeq1d 8781* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐴 = 𝐵)       (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
 
Theoremss2ixp 8782 Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)
(∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
 
Theoremixpeq2 8783 Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
(∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremixpeq2dva 8784* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremixpeq2dv 8785* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremcbvixp 8786* 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 8787* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑥 = 𝑦𝐵 = 𝐶)       X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
 
Theoremnfixpw 8788* Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8789 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2372. (Revised by Gino Giotto, 26-Jan-2024.)
𝑦𝐴    &   𝑦𝐵       𝑦X𝑥𝐴 𝐵
 
Theoremnfixp 8789 Bound-variable hypothesis builder for indexed Cartesian product. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfixpw 8788 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
𝑦𝐴    &   𝑦𝐵       𝑦X𝑥𝐴 𝐵
 
Theoremnfixp1 8790 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 8791* 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 8792* 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 8793* 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 8794* 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 8795* The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
X𝑥𝐴 (𝐵𝐶) = (X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
 
Theoremixpiin 8796* 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 8797* The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
(𝐵 ≠ ∅ → X𝑥𝐴 𝐵 = 𝑦𝐵 X𝑥𝐴 𝑦)
 
Theoremixp0x 8798 An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
X𝑥 ∈ ∅ 𝐴 = {∅}
 
Theoremixpssmap2g 8799* An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 8800 avoids ax-rep 5241. (Contributed by Mario Carneiro, 16-Nov-2014.)
( 𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴))
 
Theoremixpssmapg 8800* 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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