P. 90 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 8901-9000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fsetfocdm 8901*	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 = {𝑓 ∣ 𝑓:𝐴⟶𝐵}    &   ⊢ 𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑆:𝐹–onto→𝐵)
Theorem	fsetprcnex 8902*	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 {𝑓 ∣ 𝑓:𝐴⟶𝐵} is a set, see fsetdmprc0 8895 for 𝐴 ∉ V, fset0 8894 for 𝐴 = ∅, and fsetex 8896 for 𝐵 ∈ V, see also fsetexb 8904. (Contributed by AV, 14-Sep-2024.) (Proof shortened by BJ, 15-Sep-2024.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)
Theorem	fsetcdmex 8903*	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∈ V ↔ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V))
Theorem	fsetexb 8904*	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.)
⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V ↔ (𝐴 ∉ V ∨ 𝐴 = ∅ ∨ 𝐵 ∈ V))
Theorem	elmapfn 8905	A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐴 Fn 𝐶)
Theorem	elmapfun 8906	A mapping is always a function. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → Fun 𝐴)
Theorem	elmapssres 8907	A restricted mapping is a mapping. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷))
Theorem	fpmg 8908	A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 ∈ (𝐵 ↑pm 𝐴))
Theorem	pmss12g 8909	Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (𝐴 ↑pm 𝐵) ⊆ (𝐶 ↑pm 𝐷))
Theorem	pmresg 8910	Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))
Theorem	elmap 8911	Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐹 ∈ (𝐴 ↑m 𝐵) ↔ 𝐹:𝐵⟶𝐴)
Theorem	mapval2 8912*	Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ↑m 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵})
Theorem	elpm 8913	The predicate "is a partial function". (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (𝐵 × 𝐴)))
Theorem	elpm2 8914	The predicate "is a partial function". (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))
Theorem	fpm 8915	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:𝐴⟶𝐵 → 𝐹 ∈ (𝐵 ↑pm 𝐴))
Theorem	mapsspm 8916	Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ (𝐴 ↑m 𝐵) ⊆ (𝐴 ↑pm 𝐵)
Theorem	pmsspw 8917	Partial maps are a subset of the power set of the Cartesian product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
Theorem	mapsspw 8918	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.)
⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
Theorem	mapfvd 8919	The value of a function that maps from 𝐵 to 𝐴. (Contributed by AV, 2-Feb-2023.)
⊢ 𝑀 = (𝐴 ↑m 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐴)
Theorem	elmapresaun 8920	fresaun 6779 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)))
Theorem	fvmptmap 8921*	Special case of fvmpt 7016 for operator theorems. (Contributed by NM, 27-Nov-2007.)
⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ 𝑅 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑m 𝐷) ↦ 𝐵)    ⇒   ⊢ (𝐴:𝐷⟶𝑅 → (𝐹‘𝐴) = 𝐶)
Theorem	map0e 8922	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.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = 1o)
Theorem	map0b 8923	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) = ∅)
Theorem	map0g 8924	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑m 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
Theorem	0map0sn0 8925	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 ∅) = {∅}
Theorem	mapsnd 8926*	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})
Theorem	map0 8927	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    ⇒   ⊢ ((𝐴 ↑m 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅))
Theorem	mapsn 8928*	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    ⇒   ⊢ (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}
Theorem	mapss 8929	Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑m 𝐶) ⊆ (𝐵 ↑m 𝐶))
Theorem	fdiagfn 8930*	Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥}))    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐵⟶(𝐵 ↑m 𝐼))
Theorem	fvdiagfn 8931*	Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥}))    ⇒   ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (𝐼 × {𝑋}))
Theorem	mapsnconst 8932	Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.)
⊢ 𝑆 = {𝑋}    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑋 ∈ V    ⇒   ⊢ (𝐹 ∈ (𝐵 ↑m 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))
Theorem	mapsncnv 8933*	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 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋))    ⇒   ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
Theorem	mapsnf1o2 8934*	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 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋))    ⇒   ⊢ 𝐹:(𝐵 ↑m 𝑆)–1-1-onto→𝐵
Theorem	mapsnf1o3 8935*	Explicit bijection in the reverse of mapsnf1o2 8934. (Contributed by Stefan O'Rear, 24-Mar-2015.)
⊢ 𝑆 = {𝑋}    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑋 ∈ V    &   ⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))    ⇒   ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑m 𝑆)
Theorem	ralxpmap 8936*	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⟩}) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐽 ∈ 𝑇 → (∀𝑓 ∈ (𝑆 ↑m 𝑇)𝜑 ↔ ∀𝑦 ∈ 𝑆 ∀𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝜓))
2.4.28  Infinite Cartesian products
Syntax	cixp 8937	Extend class notation to include infinite Cartesian products.
class X𝑥 ∈ 𝐴 𝐵
Definition	df-ixp 8938*	Definition of infinite Cartesian product of [Enderton] p. 54. Enderton uses a bold "X" with 𝑥 ∈ 𝐴 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.)
⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
Theorem	dfixp 8939*	Eliminate the expression {𝑥 ∣ 𝑥 ∈ 𝐴} in df-ixp 8938, under the assumption that 𝐴 and 𝑥 are disjoint. This way, we can say that 𝑥 is bound in X𝑥 ∈ 𝐴𝐵 even if it appears free in 𝐴. (Contributed by Mario Carneiro, 12-Aug-2016.)
⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
Theorem	ixpsnval 8940*	The value of an infinite Cartesian product with a singleton. (Contributed by AV, 3-Dec-2018.)
⊢ (𝑋 ∈ 𝑉 → X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)})
Theorem	elixp2 8941*	Membership in an infinite Cartesian product. See df-ixp 8938 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)
⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
Theorem	fvixp 8942*	Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)    ⇒   ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷)
Theorem	ixpfn 8943*	A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)
⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴)
Theorem	elixp 8944*	Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
Theorem	elixpconst 8945*	Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by NM, 12-Apr-2008.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ 𝐹:𝐴⟶𝐵)
Theorem	ixpconstg 8946*	Infinite Cartesian product of a constant 𝐵. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑m 𝐴))
Theorem	ixpconst 8947*	Infinite Cartesian product of a constant 𝐵. (Contributed by NM, 28-Sep-2006.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑m 𝐴)
Theorem	ixpeq1 8948*	Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)
Theorem	ixpeq1d 8949*	Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)
Theorem	ss2ixp 8950	Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)
Theorem	ixpeq2 8951	Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)
Theorem	ixpeq2dva 8952*	Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)
Theorem	ixpeq2dv 8953*	Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)
Theorem	cbvixp 8954*	Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶
Theorem	cbvixpv 8955*	Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶
Theorem	nfixpw 8956*	Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8957 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2377. (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵
Theorem	nfixp 8957	Bound-variable hypothesis builder for indexed Cartesian product. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker nfixpw 8956 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵
Theorem	nfixp1 8958	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.)
⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵
Theorem	ixpprc 8959*	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 → X𝑥 ∈ 𝐴 𝐵 = ∅)
Theorem	ixpf 8960*	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.)
⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	uniixp 8961*	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.)
⊢ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	ixpexg 8962*	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.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ∈ V)
Theorem	ixpin 8963*	The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶)
Theorem	ixpiin 8964*	The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.)
⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶)
Theorem	ixpint 8965*	The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦)
Theorem	ixp0x 8966	An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
⊢ X𝑥 ∈ ∅ 𝐴 = {∅}
Theorem	ixpssmap2g 8967*	An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 8968 avoids ax-rep 5279. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴))
Theorem	ixpssmapg 8968*	An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴))
Theorem	0elixp 8969	Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)
⊢ ∅ ∈ X𝑥 ∈ ∅ 𝐴
Theorem	ixpn0 8970	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 10523. (Contributed by Mario Carneiro, 22-Jun-2016.)
⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅)
Theorem	ixp0 8971	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 10523. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)
⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)
Theorem	ixpssmap 8972*	An infinite Cartesian product is a subset of set exponentiation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)
⊢ 𝐵 ∈ V    ⇒   ⊢ X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴)
Theorem	resixp 8973*	Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶)
Theorem	undifixp 8974*	Union of two projections of a cartesian product. (Contributed by FL, 7-Nov-2011.)
⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ∪ 𝐺) ∈ X𝑥 ∈ 𝐴 𝐶)
Theorem	mptelixpg 8975*	Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)
⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
Theorem	resixpfo 8976*	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 = (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↦ (𝑓 ↾ 𝐵))    ⇒   ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥 ∈ 𝐴 𝐶–onto→X𝑥 ∈ 𝐵 𝐶)
Theorem	elixpsn 8977*	Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
Theorem	ixpsnf1o 8978*	A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥}))    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴)
Theorem	mapsnf1o 8979*	A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥}))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐴–1-1-onto→(𝐴 ↑m {𝐼}))
Theorem	boxriin 8980*	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.)
⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → X𝑥 ∈ 𝐼 𝐴 = (X𝑥 ∈ 𝐼 𝐵 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))
Theorem	boxcutc 8981*	The relative complement of a box set restricted on one axis. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐶 ⊆ 𝐵) → (X𝑘 ∈ 𝐴 𝐵 ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑋, 𝐶, 𝐵)) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑋, (𝐵 ∖ 𝐶), 𝐵))
2.4.29  Equinumerosity
Syntax	cen 8982	Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
class ≈
Syntax	cdom 8983	Extend class definition to include the dominance relation (curly "less than or equal to")
class ≼
Syntax	csdm 8984	Extend class definition to include the strict dominance relation (curly less-than)
class ≺
Syntax	cfn 8985	Extend class definition to include the class of all finite sets.
class Fin
Definition	df-en 8986*	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 8995. (Contributed by NM, 28-Mar-1998.)
⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
Definition	df-dom 8987*	Define the dominance relation. For an alternate definition see dfdom2 9018. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 9001 and domen 9002. (Contributed by NM, 28-Mar-1998.)
⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
Definition	df-sdom 8988	Define the strict dominance relation. Alternate possible definitions are derived as brsdom 9015 and brsdom2 9137. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
⊢ ≺ = ( ≼ ∖ ≈ )
Definition	df-fin 8989*	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 9681. If we accept Infinity, we can also express 𝐴 ∈ Fin by 𝐴 ≺ ω (Theorem isfinite 9692.) (Contributed by NM, 22-Aug-2008.)
⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
Theorem	relen 8990	Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
⊢ Rel ≈
Theorem	reldom 8991	Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
⊢ Rel ≼
Theorem	relsdom 8992	Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.)
⊢ Rel ≺
Theorem	encv 8993	If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	breng 8994*	Equinumerosity relation. This variation of bren 8995 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 8995. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
Theorem	bren 8995*	Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) Extract breng 8994 as an intermediate result. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
Theorem	brdom2g 8996*	Dominance relation. This variation of brdomg 8997 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 8997. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
Theorem	brdomg 8997*	Dominance relation. (Contributed by NM, 15-Jun-1998.) Extract brdom2g 8996 as an intermediate result. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
Theorem	brdomgOLD 8998*	Obsolete version of brdomg 8997 as of 29-Nov-2024. (Contributed by NM, 15-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
Theorem	brdomi 8999*	Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.) Avoid ax-un 7755. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
Theorem	brdomiOLD 9000*	Obsolete version of brdomi 8999 as of 29-Nov-2024. (Contributed by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)