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Theorem List for Intuitionistic Logic Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfpmg 6701 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊𝐹:𝐴𝐵) → 𝐹 ∈ (𝐵pm 𝐴))
 
Theorempmss12g 6702 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
(((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝐴pm 𝐵) ⊆ (𝐶pm 𝐷))
 
Theorempmresg 6703 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 6704 Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹 ∈ (𝐴𝑚 𝐵) ↔ 𝐹:𝐵𝐴)
 
Theoremmapval2 6705* 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 6706 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 6707 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 6708 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 6709 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 6710 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 6711 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 6712* Special case of fvmpt 5614 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 6713 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 6714 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 6715 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 6716 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 6717* 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 6718 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 6719* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝐵𝑉𝐼𝑊) → 𝐹:𝐵⟶(𝐵𝑚 𝐼))
 
Theoremfvdiagfn 6720* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝐼𝑊𝑋𝐵) → (𝐹𝑋) = (𝐼 × {𝑋}))
 
Theoremmapsnconst 6721 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 6722* 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 6723* 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 6724* Explicit bijection in the reverse of mapsnf1o2 6723. (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 6725 Extend class notation to include infinite Cartesian products.
class X𝑥𝐴 𝐵
 
Definitiondf-ixp 6726* 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 6727* Eliminate the expression {𝑥𝑥𝐴} in df-ixp 6726, 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 6728* 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 6729* Membership in an infinite Cartesian product. See df-ixp 6726 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)
(𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremfvixp 6730* Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝑥 = 𝐶𝐵 = 𝐷)       ((𝐹X𝑥𝐴 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
 
Theoremixpfn 6731* A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)
(𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
 
Theoremelixp 6732* Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
𝐹 ∈ V       (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremelixpconst 6733* Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by NM, 12-Apr-2008.)
𝐹 ∈ V       (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
 
Theoremixpconstg 6734* Infinite Cartesian product of a constant 𝐵. (Contributed by Mario Carneiro, 11-Jan-2015.)
((𝐴𝑉𝐵𝑊) → X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴))
 
Theoremixpconst 6735* Infinite Cartesian product of a constant 𝐵. (Contributed by NM, 28-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴)
 
Theoremixpeq1 6736* Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
(𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
 
Theoremixpeq1d 6737* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐴 = 𝐵)       (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
 
Theoremss2ixp 6738 Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)
(∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
 
Theoremixpeq2 6739 Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
(∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremixpeq2dva 6740* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremixpeq2dv 6741* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremcbvixp 6742* 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 6743* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑥 = 𝑦𝐵 = 𝐶)       X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
 
Theoremnfixpxy 6744* 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 6745 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 6746* 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 6747* 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 6748* 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 6749* 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)
 
Theoremixpin 6750* The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
X𝑥𝐴 (𝐵𝐶) = (X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
 
Theoremixpiinm 6751* The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
(∃𝑧 𝑧𝐵X𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 X𝑥𝐴 𝐶)
 
Theoremixpintm 6752* The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
(∃𝑧 𝑧𝐵X𝑥𝐴 𝐵 = 𝑦𝐵 X𝑥𝐴 𝑦)
 
Theoremixp0x 6753 An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
X𝑥 ∈ ∅ 𝐴 = {∅}
 
Theoremixpssmap2g 6754* An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 6755 avoids ax-coll 4133. (Contributed by Mario Carneiro, 16-Nov-2014.)
( 𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
 
Theoremixpssmapg 6755* An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.)
(∀𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
 
Theorem0elixp 6756 Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)
∅ ∈ X𝑥 ∈ ∅ 𝐴
 
Theoremixpm 6757* If an infinite Cartesian product of a family 𝐵(𝑥) is inhabited, every 𝐵(𝑥) is inhabited. (Contributed by Mario Carneiro, 22-Jun-2016.) (Revised by Jim Kingdon, 16-Feb-2023.)
(∃𝑓 𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴𝑧 𝑧𝐵)
 
Theoremixp0 6758 The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)
(∃𝑥𝐴 𝐵 = ∅ → X𝑥𝐴 𝐵 = ∅)
 
Theoremixpssmap 6759* 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 6760* 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)
 
Theoremmptelixpg 6761* Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)
(𝐼𝑉 → ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ ∀𝑥𝐼 𝐽𝐾))
 
Theoremelixpsn 6762* Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴𝑉 → (𝐹X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
 
Theoremixpsnf1o 6763* 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 6764* 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}))
 
2.6.28  Equinumerosity
 
Syntaxcen 6765 Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
class
 
Syntaxcdom 6766 Extend class definition to include the dominance relation (curly less-than-or-equal)
class
 
Syntaxcfn 6767 Extend class definition to include the class of all finite sets.
class Fin
 
Definitiondf-en 6768* 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 6774. (Contributed by NM, 28-Mar-1998.)
≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
 
Definitiondf-dom 6769* Define the dominance relation. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6777 and domen 6778. (Contributed by NM, 28-Mar-1998.)
≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
 
Definitiondf-fin 6770* Define the (proper) class of all finite sets. Similar to Definition 10.29 of [TakeutiZaring] p. 91, whose "Fin(a)" corresponds to our "𝑎 ∈ Fin". This definition is meaningful whether or not we accept the Axiom of Infinity ax-inf2 15206. (Contributed by NM, 22-Aug-2008.)
Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦}
 
Theoremrelen 6771 Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
Rel ≈
 
Theoremreldom 6772 Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
Rel ≼
 
Theoremencv 6773 If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
(𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theorembren 6774* Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
(𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
 
Theorembrdomg 6775* Dominance relation. (Contributed by NM, 15-Jun-1998.)
(𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
 
Theorembrdomi 6776* Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
(𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
 
Theorembrdom 6777* Dominance relation. (Contributed by NM, 15-Jun-1998.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵)
 
Theoremdomen 6778* Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
 
Theoremdomeng 6779* Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.)
(𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
 
Theoremctex 6780 A class dominated by ω is a set. See also ctfoex 7148 which says that a countable class is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.)
(𝐴 ≼ ω → 𝐴 ∈ V)
 
Theoremf1oen3g 6781 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6784 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
 
Theoremf1oen2g 6782 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6784 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
 
Theoremf1dom2g 6783 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6785 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)
 
Theoremf1oeng 6784 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
((𝐴𝐶𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
 
Theoremf1domg 6785 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)
(𝐵𝐶 → (𝐹:𝐴1-1𝐵𝐴𝐵))
 
Theoremf1oen 6786 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
𝐴 ∈ V       (𝐹:𝐴1-1-onto𝐵𝐴𝐵)
 
Theoremf1dom 6787 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.)
𝐵 ∈ V       (𝐹:𝐴1-1𝐵𝐴𝐵)
 
Theoremisfi 6788* Express "𝐴 is finite". Definition 10.29 of [TakeutiZaring] p. 91 (whose "Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)
(𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
 
Theoremenssdom 6789 Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
≈ ⊆ ≼
 
Theoremendom 6790 Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
(𝐴𝐵𝐴𝐵)
 
Theoremenrefg 6791 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴𝑉𝐴𝐴)
 
Theoremenref 6792 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
𝐴 ∈ V       𝐴𝐴
 
Theoremeqeng 6793 Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
(𝐴𝑉 → (𝐴 = 𝐵𝐴𝐵))
 
Theoremdomrefg 6794 Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
(𝐴𝑉𝐴𝐴)
 
Theoremen2d 6795* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑 → (𝑥𝐴𝐶 ∈ V))    &   (𝜑 → (𝑦𝐵𝐷 ∈ V))    &   (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))       (𝜑𝐴𝐵)
 
Theoremen3d 6796* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑 → (𝑥𝐴𝐶𝐵))    &   (𝜑 → (𝑦𝐵𝐷𝐴))    &   (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶)))       (𝜑𝐴𝐵)
 
Theoremen2i 6797* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥𝐴𝐶 ∈ V)    &   (𝑦𝐵𝐷 ∈ V)    &   ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷))       𝐴𝐵
 
Theoremen3i 6798* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥𝐴𝐶𝐵)    &   (𝑦𝐵𝐷𝐴)    &   ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶))       𝐴𝐵
 
Theoremdom2lem 6799* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)
(𝜑 → (𝑥𝐴𝐶𝐵))    &   (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))       (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)
 
Theoremdom2d 6800* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.)
(𝜑 → (𝑥𝐴𝐶𝐵))    &   (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))       (𝜑 → (𝐵𝑅𝐴𝐵))
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