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Theorem List for Metamath Proof Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-map 7901* Define the mapping operation or set exponentiation. The set of all functions that map from 𝐵 to 𝐴 is written (𝐴𝑚 𝐵) (see mapval 7911). 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 7902* 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 7910). 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 7901) . See mapsspm 7933 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)
pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
 
Theoremmapprc 7903* 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 7904* 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 7905* 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 7906 Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
𝑚 Fn (V × V)
 
Theoremfnpm 7907 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
pm Fn (V × V)
 
Theoremreldmmap 7908 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Rel dom ↑𝑚
 
Theoremmapvalg 7909* 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 7910* 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 7911* 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 7912 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 7913 Deduction form of elmapg 7912. (Contributed by BJ, 11-Apr-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐶 ∈ (𝐴𝑚 𝐵) ↔ 𝐶:𝐵𝐴))
 
Theoremmapdm0 7914 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 7915 The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)
((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴pm 𝐵) ↔ (Fun 𝐶𝐶 ⊆ (𝐵 × 𝐴))))
 
Theoremelpm2g 7916 The predicate "is a partial function." (Contributed by NM, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊) → (𝐹 ∈ (𝐴pm 𝐵) ↔ (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵)))
 
Theoremelpm2r 7917 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐹:𝐶𝐴𝐶𝐵)) → 𝐹 ∈ (𝐴pm 𝐵))
 
Theoremelpmi 7918 A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)
(𝐹 ∈ (𝐴pm 𝐵) → (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵))
 
Theorempmfun 7919 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 7920 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 7921 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 7922 A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
(𝐴 ∈ (𝐵𝑚 𝐶) → 𝐴 Fn 𝐶)
 
Theoremelmapfun 7923 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 7924 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 7925 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊𝐹:𝐴𝐵) → 𝐹 ∈ (𝐵pm 𝐴))
 
Theorempmss12g 7926 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
(((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝐴pm 𝐵) ⊆ (𝐶pm 𝐷))
 
Theorempmresg 7927 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 7928 Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹 ∈ (𝐴𝑚 𝐵) ↔ 𝐹:𝐵𝐴)
 
Theoremmapval2 7929* 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 7930 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 7931 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 7932 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 7933 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 7934 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 7935 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 7936* Special case of fvmpt 6321 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 7937 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 7938 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 7939 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 7940 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 7941* 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 7942 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 7943* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝐵𝑉𝐼𝑊) → 𝐹:𝐵⟶(𝐵𝑚 𝐼))
 
Theoremfvdiagfn 7944* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝐼𝑊𝑋𝐵) → (𝐹𝑋) = (𝐼 × {𝑋}))
 
Theoremmapsnconst 7945 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 7946* 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 7947* 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 7948* Explicit bijection in the reverse of mapsnf1o2 7947. (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 7949* 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 7950 Extend class notation to include infinite Cartesian products.
class X𝑥𝐴 𝐵
 
Definitiondf-ixp 7951* 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 7952* Eliminate the expression {𝑥𝑥𝐴} in df-ixp 7951, 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 7953* 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 7954* Membership in an infinite Cartesian product. See df-ixp 7951 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)
(𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremfvixp 7955* Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝑥 = 𝐶𝐵 = 𝐷)       ((𝐹X𝑥𝐴 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
 
Theoremixpfn 7956* A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)
(𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
 
Theoremelixp 7957* Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
𝐹 ∈ V       (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremelixpconst 7958* Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by NM, 12-Apr-2008.)
𝐹 ∈ V       (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
 
Theoremixpconstg 7959* Infinite Cartesian product of a constant 𝐵. (Contributed by Mario Carneiro, 11-Jan-2015.)
((𝐴𝑉𝐵𝑊) → X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴))
 
Theoremixpconst 7960* Infinite Cartesian product of a constant 𝐵. (Contributed by NM, 28-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴)
 
Theoremixpeq1 7961* Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
(𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
 
Theoremixpeq1d 7962* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐴 = 𝐵)       (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
 
Theoremss2ixp 7963 Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)
(∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
 
Theoremixpeq2 7964 Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
(∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremixpeq2dva 7965* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremixpeq2dv 7966* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremcbvixp 7967* 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 7968* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑥 = 𝑦𝐵 = 𝐶)       X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
 
Theoremnfixp 7969 Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.)
𝑦𝐴    &   𝑦𝐵       𝑦X𝑥𝐴 𝐵
 
Theoremnfixp1 7970 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 7971* 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 7972* 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 7973* 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 7974* 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 7975* The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
X𝑥𝐴 (𝐵𝐶) = (X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
 
Theoremixpiin 7976* 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 7977* The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
(𝐵 ≠ ∅ → X𝑥𝐴 𝐵 = 𝑦𝐵 X𝑥𝐴 𝑦)
 
Theoremixp0x 7978 An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
X𝑥 ∈ ∅ 𝐴 = {∅}
 
Theoremixpssmap2g 7979* An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 7980 avoids ax-rep 4804. (Contributed by Mario Carneiro, 16-Nov-2014.)
( 𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
 
Theoremixpssmapg 7980* An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.)
(∀𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
 
Theorem0elixp 7981 Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)
∅ ∈ X𝑥 ∈ ∅ 𝐴
 
Theoremixpn0 7982 The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 9343. (Contributed by Mario Carneiro, 22-Jun-2016.)
(X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)
 
Theoremixp0 7983 The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 9343. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)
(∃𝑥𝐴 𝐵 = ∅ → X𝑥𝐴 𝐵 = ∅)
 
Theoremixpssmap 7984* An infinite Cartesian product is a subset of set exponentiation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)
𝐵 ∈ V       X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴)
 
Theoremresixp 7985* Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
((𝐵𝐴𝐹X𝑥𝐴 𝐶) → (𝐹𝐵) ∈ X𝑥𝐵 𝐶)
 
Theoremundifixp 7986* Union of two projections of a cartesian product. (Contributed by FL, 7-Nov-2011.)
((𝐹X𝑥𝐵 𝐶𝐺X𝑥 ∈ (𝐴𝐵)𝐶𝐵𝐴) → (𝐹𝐺) ∈ X𝑥𝐴 𝐶)
 
Theoremmptelixpg 7987* Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)
(𝐼𝑉 → ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ ∀𝑥𝐼 𝐽𝐾))
 
Theoremresixpfo 7988* Restriction of elements of an infinite Cartesian product creates a surjection, if the original Cartesian product is nonempty. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝐹 = (𝑓X𝑥𝐴 𝐶 ↦ (𝑓𝐵))       ((𝐵𝐴X𝑥𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥𝐴 𝐶ontoX𝑥𝐵 𝐶)
 
Theoremelixpsn 7989* Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴𝑉 → (𝐹X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
 
Theoremixpsnf1o 7990* A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐴 ↦ ({𝐼} × {𝑥}))       (𝐼𝑉𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴)
 
Theoremmapsnf1o 7991* A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐴 ↦ ({𝐼} × {𝑥}))       ((𝐴𝑉𝐼𝑊) → 𝐹:𝐴1-1-onto→(𝐴𝑚 {𝐼}))
 
Theoremboxriin 7992* A rectangular subset of a rectangular set can be recovered as the relative intersection of single-axis restrictions. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(∀𝑥𝐼 𝐴𝐵X𝑥𝐼 𝐴 = (X𝑥𝐼 𝐵 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))
 
Theoremboxcutc 7993* The relative complement of a box set restricted on one axis. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝑋𝐴 ∧ ∀𝑘𝐴 𝐶𝐵) → (X𝑘𝐴 𝐵X𝑘𝐴 if(𝑘 = 𝑋, 𝐶, 𝐵)) = X𝑘𝐴 if(𝑘 = 𝑋, (𝐵𝐶), 𝐵))
 
2.4.23  Equinumerosity
 
Syntaxcen 7994 Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
class
 
Syntaxcdom 7995 Extend class definition to include the dominance relation (curly less-than-or-equal)
class
 
Syntaxcsdm 7996 Extend class definition to include the strict dominance relation (curly less-than)
class
 
Syntaxcfn 7997 Extend class definition to include the class of all finite sets.
class Fin
 
Definitiondf-en 7998* Define the equinumerosity relation. Definition of [Enderton] p. 129. We define to be a binary relation rather than a connective, so its arguments must be sets to be meaningful. This is acceptable because we do not consider equinumerosity for proper classes. We derive the usual definition as bren 8006. (Contributed by NM, 28-Mar-1998.)
≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
 
Definitiondf-dom 7999* Define the dominance relation. For an alternate definition see dfdom2 8023. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 8009 and domen 8010. (Contributed by NM, 28-Mar-1998.)
≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
 
Definitiondf-sdom 8000 Define the strict dominance relation. Alternate possible definitions are derived as brsdom 8020 and brsdom2 8125. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
≺ = ( ≼ ∖ ≈ )
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