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Theorem List for Metamath Proof Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremunbnn 8101* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. Part of the proof of Theorem 42 of [Suppes] p. 151. See unbnn3 8439 for a stronger version without the first assumption. (Contributed by NM, 3-Dec-2003.)
((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)

Theoremunbnn2 8102* Version of unbnn 8101 that does not require a strict upper bound. (Contributed by NM, 24-Apr-2004.)
((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)

Theoremisfinite2 8103 Any set strictly dominated by the class of natural numbers is finite. Sufficiency part of Theorem 42 of [Suppes] p. 151. This theorem does not require the Axiom of Infinity. (Contributed by NM, 24-Apr-2004.)
(𝐴 ≺ ω → 𝐴 ∈ Fin)

Theoremnnsdomg 8104 Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 15-Jun-1998.)
((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)

Theoremisfiniteg 8105 A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 3-Nov-2002.) (Revised by Mario Carneiro, 27-Apr-2015.)
(ω ∈ V → (𝐴 ∈ Fin ↔ 𝐴 ≺ ω))

Theoreminfsdomnn 8106 An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
((ω ≼ 𝐴𝐵 ∈ ω) → 𝐵𝐴)

Theoreminfn0 8107 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
(ω ≼ 𝐴𝐴 ≠ ∅)

Theoremfin2inf 8108 This (useless) theorem, which was proved without the Axiom of Infinity, demonstrates an artifact of our definition of binary relation, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless ω exists. (Contributed by NM, 13-Nov-2003.)
(𝐴 ≺ ω → ω ∈ V)

Theoremunfilem1 8109* Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ ω    &   𝐵 ∈ ω    &   𝐹 = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))       ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)

Theoremunfilem2 8110* Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ ω    &   𝐵 ∈ ω    &   𝐹 = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))       𝐹:𝐵1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)

Theoremunfilem3 8111 Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 16-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 ≈ ((𝐴 +𝑜 𝐵) ∖ 𝐴))

Theoremunfi 8112 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)

Theoremunfir 8113 If a union is finite, the operands are finite. Converse of unfi 8112. (Contributed by FL, 3-Aug-2009.)
((𝐴𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))

Theoremunfi2 8114 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 8112 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 8108). (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)

Theoremdifinf 8115 An infinite set 𝐴 minus a finite set is infinite. (Contributed by FL, 3-Aug-2009.)
((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ¬ (𝐴𝐵) ∈ Fin)

Theoremxpfi 8116 The Cartesian product of two finite sets is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)

Theorem3xpfi 8117 The Cartesian product of three finite sets is a finite set. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
(𝑉 ∈ Fin → ((𝑉 × 𝑉) × 𝑉) ∈ Fin)

Theoremdomunfican 8118 A finite set union cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
(((𝐴 ∈ Fin ∧ 𝐵𝐴) ∧ ((𝐴𝑋) = ∅ ∧ (𝐵𝑌) = ∅)) → ((𝐴𝑋) ≼ (𝐵𝑌) ↔ 𝑋𝑌))

Theoreminfcntss 8119* Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.) (Contributed by NM, 23-Oct-2004.)
𝐴 ∈ V       (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ ω))

Theoremprfi 8120 An unordered pair is finite. (Contributed by NM, 22-Aug-2008.)
{𝐴, 𝐵} ∈ Fin

Theoremtpfi 8121 An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.)
{𝐴, 𝐵, 𝐶} ∈ Fin

Theoremfiint 8122* Equivalent ways of stating the finite intersection property. We show two ways of saying, "the intersection of elements in every finite nonempty subcollection of 𝐴 is in 𝐴." This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally. (Contributed by NM, 22-Sep-2002.)
(∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝐴))

Theoremfnfi 8123 A version of fnex 6386 for finite sets that does not require Replacement. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
((𝐹 Fn 𝐴𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Theoremfodomfi 8124 An onto function implies dominance of domain over range, for finite sets. Unlike fodom 9227 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → 𝐵𝐴)

Theoremfodomfib 8125* Equivalence of an onto mapping and dominance for a nonempty finite set. Unlike fodomb 9229 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.)
(𝐴 ∈ Fin → ((𝐴 ≠ ∅ ∧ ∃𝑓 𝑓:𝐴onto𝐵) ↔ (∅ ≺ 𝐵𝐵𝐴)))

Theoremfofinf1o 8126 Any surjection from one finite set to another of equal size must be a bijection. (Contributed by Mario Carneiro, 19-Aug-2014.)
((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1-onto𝐵)

Theoremrneqdmfinf1o 8127 Any function from a finite set onto the same set must be a bijection. (Contributed by AV, 5-Jul-2021.)
((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴1-1-onto𝐴)

Theoremfidomdm 8128 Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
(𝐹 ∈ Fin → dom 𝐹𝐹)

Theoremdmfi 8129 The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)
(𝐴 ∈ Fin → dom 𝐴 ∈ Fin)

Theoremfundmfibi 8130 A function (set) is finite if and only if its domain is finite. (Contributed by AV, 10-Jan-2020.)
(Fun 𝐹 → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))

Theoremcnvfi 8131 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ∈ Fin → 𝐴 ∈ Fin)

Theoremrnfi 8132 The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ∈ Fin → ran 𝐴 ∈ Fin)

Theoremf1dmvrnfibi 8133 A 1-1 function (class) with a set as domain is finite if and only if its range is finite. (Contributed by AV, 10-Jan-2020.)
((𝐴𝑉𝐹:𝐴1-1𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

Theoremf1vrnfibi 8134 A 1-1 function (set) is finite if and only if its range is finite. (Contributed by AV, 10-Jan-2020.)
((𝐹𝑉𝐹:𝐴1-1𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

Theoremfofi 8135 If a function has a finite domain, its range is finite. Theorem 37 of [Suppes] p. 104. (Contributed by NM, 25-Mar-2007.)
((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → 𝐵 ∈ Fin)

Theoremf1fi 8136 If a 1-to-1 function has a finite codomain its domain is finite. (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 24-Jun-2015.)
((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐴 ∈ Fin)

Theoremiunfi 8137* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 8138. Note that 𝐵 depends on 𝑥, i.e. can be thought of as 𝐵(𝑥). (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
((𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ Fin) → 𝑥𝐴 𝐵 ∈ Fin)

Theoremunifi 8138 The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. (Contributed by NM, 22-Aug-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)

Theoremunifi2 8139* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 8138 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 8108). (Contributed by NM, 11-Mar-2006.)
((𝐴 ≺ ω ∧ ∀𝑥𝐴 𝑥 ≺ ω) → 𝐴 ≺ ω)

Theoreminfssuni 8140* If an infinite set 𝐴 is included in the underlying set of a finite cover 𝐵, then there exists a set of the cover that contains an infinite number of element of 𝐴. (Contributed by FL, 2-Aug-2009.)
((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐴 𝐵) → ∃𝑥𝐵 ¬ (𝐴𝑥) ∈ Fin)

Theoremunirnffid 8141 The union of the range of a function from a finite set into the class of finite sets is finite. Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐹:𝑇⟶Fin)    &   (𝜑𝑇 ∈ Fin)       (𝜑 ran 𝐹 ∈ Fin)

Theoremimafi 8142 Images of finite sets are finite. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((Fun 𝐹𝑋 ∈ Fin) → (𝐹𝑋) ∈ Fin)

Theorempwfilem 8143* Lemma for pwfi 8144. (Contributed by NM, 26-Mar-2007.)
𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))       (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)

Theorempwfi 8144 The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105. (Contributed by NM, 26-Mar-2007.)
(𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)

Theoremmapfi 8145 Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝑚 𝐵) ∈ Fin)

Theoremixpfi 8146* A Cartesian product of finitely many finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ Fin) → X𝑥𝐴 𝐵 ∈ Fin)

Theoremixpfi2 8147* A Cartesian product of finite sets such that all but finitely many are singletons is finite. (Note that 𝐵(𝑥) and 𝐷(𝑥) are both possibly dependent on 𝑥.) (Contributed by Mario Carneiro, 25-Jan-2015.)
(𝜑𝐶 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)    &   ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐵 ⊆ {𝐷})       (𝜑X𝑥𝐴 𝐵 ∈ Fin)

Theoremmptfi 8148* A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴 ∈ Fin → (𝑥𝐴𝐵) ∈ Fin)

Theoremabrexfi 8149* An image set from a finite set is finite. (Contributed by Mario Carneiro, 13-Feb-2014.)
(𝐴 ∈ Fin → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ Fin)

Theoremcnvimamptfin 8150* A preimage of a mapping with a finite domain under any class is finite. In contrast to fisuppfi 8166, the range of the mapping needs not to be known. (Contributed by AV, 21-Dec-2018.)
(𝜑𝑁 ∈ Fin)       (𝜑 → ((𝑝𝑁𝑋) “ 𝑌) ∈ Fin)

Theoremelfpw 8151 Membership in a class of finite subsets. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝐴 ∈ (𝒫 𝐵 ∩ Fin) ↔ (𝐴𝐵𝐴 ∈ Fin))

Theoremunifpw 8152 A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝒫 𝐴 ∩ Fin) = 𝐴

Theoremf1opwfi 8153* A one-to-one mapping induces a one-to-one mapping on finite subsets. (Contributed by Mario Carneiro, 25-Jan-2015.)
(𝐹:𝐴1-1-onto𝐵 → (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐹𝑏)):(𝒫 𝐴 ∩ Fin)–1-1-onto→(𝒫 𝐵 ∩ Fin))

Theoremfissuni 8154* A finite subset of a union is covered by finitely many elements. (Contributed by Stefan O'Rear, 2-Apr-2015.)
((𝐴 𝐵𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 𝑐)

Theoremfipreima 8155* Given a finite subset 𝐴 of the range of a function, there exists a finite subset of the domain whose image is 𝐴. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 22-Feb-2015.)
((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)

Theoremfinsschain 8156* A finite subset of the union of a superset chain is a subset of some element of the chain. A useful preliminary result for alexsub 21659 and others. (Contributed by Jeff Hankins, 25-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 18-May-2015.)
(((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝐵 ∈ Fin ∧ 𝐵 𝐴)) → ∃𝑧𝐴 𝐵𝑧)

Theoremindexfi 8157* If for every element of a finite indexing set 𝐴 there exists a corresponding element of another set 𝐵, then there exists a finite subset of 𝐵 consisting only of those elements which are indexed by 𝐴. Proven without the Axiom of Choice, unlike indexdom 32699. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝐴 ∈ Fin ∧ 𝐵𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑐 ∈ Fin (𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))

2.4.28  Finitely supported functions

Syntaxcfsupp 8158 Extend class definition to include the predicate to be a finitely supported function.
class finSupp

Definitiondf-fsupp 8159* Define the property of a function to be finitely supported (in relation to a given zero). (Contributed by AV, 23-May-2019.)
finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}

Theoremrelfsupp 8160 The property of a function to be finitely supported is a relation. (Contributed by AV, 7-Jun-2019.)
Rel finSupp

Theoremrelprcnfsupp 8161 A proper class is never finitely supported. (Contributed by AV, 7-Jun-2019.)
𝐴 ∈ V → ¬ 𝐴 finSupp 𝑍)

Theoremisfsupp 8162 The property of a class to be a finitely supported function (in relation to a given zero). (Contributed by AV, 23-May-2019.)
((𝑅𝑉𝑍𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))

Theoremfunisfsupp 8163 The property of a function to be finitely supported (in relation to a given zero). (Contributed by AV, 23-May-2019.)
((Fun 𝑅𝑅𝑉𝑍𝑊) → (𝑅 finSupp 𝑍 ↔ (𝑅 supp 𝑍) ∈ Fin))

Theoremfsuppimp 8164 Implications of a class being a finitely supported function (in relation to a given zero). (Contributed by AV, 26-May-2019.)
(𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))

Theoremfsuppimpd 8165 A finitely supported function is a function with a finite support. (Contributed by AV, 6-Jun-2019.)
(𝜑𝐹 finSupp 𝑍)       (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Theoremfisuppfi 8166 A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝐶) ∈ Fin)

Theoremfdmfisuppfi 8167 The support of a function with a finite domain is always finite. (Contributed by AV, 27-Apr-2019.)
(𝜑𝐹:𝐷𝑅)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑍𝑉)       (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Theoremfdmfifsupp 8168 A function with a finite domain is always finitely supported. (Contributed by AV, 25-May-2019.)
(𝜑𝐹:𝐷𝑅)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑍𝑉)       (𝜑𝐹 finSupp 𝑍)

Theoremfsuppmptdm 8169* A mapping with a finite domain is finitely supported. (Contributed by AV, 7-Jun-2019.)
𝐹 = (𝑥𝐴𝑌)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝑌𝑉)    &   (𝜑𝑍𝑊)       (𝜑𝐹 finSupp 𝑍)

Theoremfndmfisuppfi 8170 The support of a function with a finite domain is always finite. (Contributed by AV, 25-May-2019.)
(𝜑𝐹 Fn 𝐷)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑍𝑉)       (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Theoremfndmfifsupp 8171 A function with a finite domain is always finitely supported. (Contributed by AV, 25-May-2019.)
(𝜑𝐹 Fn 𝐷)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑍𝑉)       (𝜑𝐹 finSupp 𝑍)

Theoremsuppeqfsuppbi 8172 If two functions have the same support, one function is finitely supported iff the other one is finitely supported. (Contributed by AV, 30-Jun-2019.)
(((𝐹𝑈 ∧ Fun 𝐹) ∧ (𝐺𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍𝐺 finSupp 𝑍)))

Theoremsuppssfifsupp 8173 If the support of a function is a subset of a finite set, the function is finitely supported. (Contributed by AV, 15-Jul-2019.)
(((𝐺𝑉 ∧ Fun 𝐺𝑍𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → 𝐺 finSupp 𝑍)

Theoremfsuppsssupp 8174 If the support of a function is a subset of the support of a finitely supported function, the function is finitely supported. (Contributed by AV, 2-Jul-2019.) (Proof shortened by AV, 15-Jul-2019.)
(((𝐺𝑉 ∧ Fun 𝐺) ∧ (𝐹 finSupp 𝑍 ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))) → 𝐺 finSupp 𝑍)

Theoremfsuppxpfi 8175 The cartesian product of two finitely supported functions is finite. (Contributed by AV, 17-Jul-2019.)
((𝐹 finSupp 𝑍𝐺 finSupp 𝑍) → ((𝐹 supp 𝑍) × (𝐺 supp 𝑍)) ∈ Fin)

Theoremfczfsuppd 8176 A constant function with value zero is finitely supported. (Contributed by AV, 30-Jun-2019.)
(𝜑𝐵𝑉)    &   (𝜑𝑍𝑊)       (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍)

Theoremfsuppun 8177 The union of two finitely supported functions is finitely supported (but not necessarily a function!). (Contributed by AV, 3-Jun-2019.)
(𝜑𝐹 finSupp 𝑍)    &   (𝜑𝐺 finSupp 𝑍)       (𝜑 → ((𝐹𝐺) supp 𝑍) ∈ Fin)

Theoremfsuppunfi 8178 The union of the support of two finitely supported functions is finite. (Contributed by AV, 1-Jul-2019.)
(𝜑𝐹 finSupp 𝑍)    &   (𝜑𝐺 finSupp 𝑍)       (𝜑 → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin)

Theoremfsuppunbi 8179 If the union of two classes/functions is a function, this union is finitely supported iff the two functions are finitely supported. (Contributed by AV, 18-Jun-2019.)
(𝜑 → Fun (𝐹𝐺))       (𝜑 → ((𝐹𝐺) finSupp 𝑍 ↔ (𝐹 finSupp 𝑍𝐺 finSupp 𝑍)))

Theorem0fsupp 8180 The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
(𝑍𝑉 → ∅ finSupp 𝑍)

Theoremsnopfsupp 8181 A singleton containing an ordered pair is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩} finSupp 𝑍)

Theoremfunsnfsupp 8182 Finite support for a function extended by a singleton. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by AV, 19-Jul-2019.)
(((𝑋𝑉𝑌𝑊) ∧ (Fun 𝐹𝑋 ∉ dom 𝐹)) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) finSupp 𝑍𝐹 finSupp 𝑍))

Theoremfsuppres 8183 The restriction of a finitely supported function is finitely supported. (Contributed by AV, 14-Jul-2019.)
(𝜑𝐹 finSupp 𝑍)    &   (𝜑𝑍𝑉)       (𝜑 → (𝐹𝑋) finSupp 𝑍)

Theoremressuppfi 8184 If the support of the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finite, the support of the function itself is finite. (Contributed by AV, 22-Apr-2019.)
(𝜑 → (dom 𝐹𝐵) ∈ Fin)    &   (𝜑𝐹𝑊)    &   (𝜑𝐺 = (𝐹𝐵))    &   (𝜑 → (𝐺 supp 𝑍) ∈ Fin)    &   (𝜑𝑍𝑉)       (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Theoremresfsupp 8185 If the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finitely supported, the function itself is finitely supported. (Contributed by AV, 27-May-2019.)
(𝜑 → (dom 𝐹𝐵) ∈ Fin)    &   (𝜑𝐹𝑊)    &   (𝜑 → Fun 𝐹)    &   (𝜑𝐺 = (𝐹𝐵))    &   (𝜑𝐺 finSupp 𝑍)    &   (𝜑𝑍𝑉)       (𝜑𝐹 finSupp 𝑍)

Theoremresfifsupp 8186 The restriction of a function to a finite set is finitely supported. (Contributed by AV, 12-Dec-2019.)
(𝜑 → Fun 𝐹)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑍𝑉)       (𝜑 → (𝐹𝑋) finSupp 𝑍)

Theoremfrnfsuppbi 8187 Two ways of saying that a function with known codomain is finitely supported. (Contributed by AV, 8-Jul-2019.)
((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 finSupp 𝑍 ↔ (𝐹 “ (𝑆 ∖ {𝑍})) ∈ Fin)))

Theoremfsuppmptif 8188* A function mapping an argument to either a value of a finitely supported function or zero is finitely supported. (Contributed by AV, 6-Jun-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝑍𝑊)    &   (𝜑𝐹 finSupp 𝑍)       (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 𝑍)) finSupp 𝑍)

Theoremfsuppcolem 8189 Lemma for fsuppco 8190. Formula building theorem for finite supports: rearranging the index set. (Contributed by Stefan O'Rear, 21-Mar-2015.)
(𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ Fin)    &   (𝜑𝐺:𝑋1-1𝑌)       (𝜑 → ((𝐹𝐺) “ (V ∖ {𝑍})) ∈ Fin)

Theoremfsuppco 8190 The composition of a 1-1 function with a finitely supported function is finitely supported. (Contributed by AV, 28-May-2019.)
(𝜑𝐹 finSupp 𝑍)    &   (𝜑𝐺:𝑋1-1𝑌)    &   (𝜑𝑍𝑊)    &   (𝜑𝐹𝑉)       (𝜑 → (𝐹𝐺) finSupp 𝑍)

Theoremfsuppco2 8191 The composition of a function which maps the zero to zero with a finitely supported function is finitely supported. This is not only a special case of fsuppcor 8192 because it does not require that the "zero" is an element of the range of the finitely supported function. (Contributed by AV, 6-Jun-2019.)
(𝜑𝑍𝑊)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐵𝐵)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐹 finSupp 𝑍)    &   (𝜑 → (𝐺𝑍) = 𝑍)       (𝜑 → (𝐺𝐹) finSupp 𝑍)

Theoremfsuppcor 8192 The composition of a function which maps the zero of the range of a finitely supported function to the zero of its range with this finitely supported function is finitely supported. (Contributed by AV, 6-Jun-2019.)
(𝜑0𝑊)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹:𝐴𝐶)    &   (𝜑𝐺:𝐵𝐷)    &   (𝜑𝐶𝐵)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐹 finSupp 𝑍)    &   (𝜑 → (𝐺𝑍) = 0 )       (𝜑 → (𝐺𝐹) finSupp 0 )

Theoremmapfienlem1 8193* Lemma 1 for mapfien 8196. (Contributed by AV, 3-Jul-2019.)
𝑆 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷𝑚 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺𝑍)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐷 ∈ V)    &   (𝜑𝑍𝐵)       ((𝜑𝑓𝑆) → (𝐺 ∘ (𝑓𝐹)) finSupp 𝑊)

Theoremmapfienlem2 8194* Lemma 2 for mapfien 8196. (Contributed by AV, 3-Jul-2019.)
𝑆 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷𝑚 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺𝑍)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐷 ∈ V)    &   (𝜑𝑍𝐵)       ((𝜑𝑔𝑇) → ((𝐺𝑔) ∘ 𝐹) finSupp 𝑍)

Theoremmapfienlem3 8195* Lemma 3 for mapfien 8196. (Contributed by AV, 3-Jul-2019.)
𝑆 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷𝑚 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺𝑍)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐷 ∈ V)    &   (𝜑𝑍𝐵)       ((𝜑𝑔𝑇) → ((𝐺𝑔) ∘ 𝐹) ∈ 𝑆)

Theoremmapfien 8196* A bijection of the base sets induces a bijection on the set of finitely supported functions. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
𝑆 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷𝑚 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺𝑍)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐷 ∈ V)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑓𝑆 ↦ (𝐺 ∘ (𝑓𝐹))):𝑆1-1-onto𝑇)

Theoremmapfien2 8197* Equinumerousity relation for sets of finitely supported functions. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
𝑆 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 0 }    &   𝑇 = {𝑥 ∈ (𝐷𝑚 𝐶) ∣ 𝑥 finSupp 𝑊}    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑0𝐵)    &   (𝜑𝑊𝐷)       (𝜑𝑆𝑇)

Theoremsniffsupp 8198* A function mapping all but one arguments to zero is finitely supported. (Contributed by AV, 8-Jul-2019.)
(𝜑𝐼𝑉)    &   (𝜑0𝑊)    &   𝐹 = (𝑥𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 ))       (𝜑𝐹 finSupp 0 )

2.4.29  Finite intersections

Syntaxcfi 8199 Extend class notation with the function whose value is the class of all the finite intersections of the elements of a given set.
class fi

Definitiondf-fi 8200* Function whose value is the class of all the finite intersections of the elements of 𝑥. (Contributed by FL, 27-Apr-2008.)
fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = 𝑦})

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