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Theorem List for Metamath Proof Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfodomfi 9101 An onto function implies dominance of domain over range, for finite sets. Unlike fodom 10288 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 9102* Equivalence of an onto mapping and dominance for a nonempty finite set. Unlike fodomb 10291 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 9103 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 9104 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 9105 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 9106 The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)
(𝐴 ∈ Fin → dom 𝐴 ∈ Fin)
 
Theoremfundmfibi 9107 A function is finite if and only if its domain is finite. (Contributed by AV, 10-Jan-2020.)
(Fun 𝐹 → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
 
Theoremresfnfinfin 9108 The restriction of a function to a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹𝐵) ∈ Fin)
 
Theoremresidfi 9109 A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)
(( I ↾ 𝐴) ∈ Fin ↔ 𝐴 ∈ Fin)
 
TheoremcnvfiALT 9110 Shorter proof of cnvfi 8972 using ax-pow 5289. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ Fin → 𝐴 ∈ Fin)
 
Theoremrnfi 9111 The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
 
Theoremf1dmvrnfibi 9112 A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 9113. (Contributed by AV, 10-Jan-2020.)
((𝐴𝑉𝐹:𝐴1-1𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
 
Theoremf1vrnfibi 9113 A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 9112. (Contributed by AV, 10-Jan-2020.)
((𝐹𝑉𝐹:𝐴1-1𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
 
Theoremfofi 9114 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 9115 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 9116* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 9117. 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 9117 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 9118* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 9117 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 9086). (Contributed by NM, 11-Mar-2006.)
((𝐴 ≺ ω ∧ ∀𝑥𝐴 𝑥 ≺ ω) → 𝐴 ≺ ω)
 
Theoreminfssuni 9119* 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 9120 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)
 
TheoremimafiALT 9121 Shorter proof of imafi 8967 using ax-pow 5289. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
((Fun 𝐹𝑋 ∈ Fin) → (𝐹𝑋) ∈ Fin)
 
TheorempwfilemOLD 9122* Obsolete version of pwfilem 8969 as of 7-Sep-2024. (Contributed by NM, 26-Mar-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))       (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
 
TheorempwfiOLD 9123 Obsolete version of pwfi 8970 as of 7-Sep-2024. (Contributed by NM, 26-Mar-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
 
Theoremmapfi 9124 Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴m 𝐵) ∈ Fin)
 
Theoremixpfi 9125* 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 9126* 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 9127* A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴 ∈ Fin → (𝑥𝐴𝐵) ∈ Fin)
 
Theoremabrexfi 9128* An image set from a finite set is finite. (Contributed by Mario Carneiro, 13-Feb-2014.)
(𝐴 ∈ Fin → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ Fin)
 
Theoremcnvimamptfin 9129* A preimage of a mapping with a finite domain under any class is finite. In contrast to fisuppfi 9145, the range of the mapping needs not to be known. (Contributed by AV, 21-Dec-2018.)
(𝜑𝑁 ∈ Fin)       (𝜑 → ((𝑝𝑁𝑋) “ 𝑌) ∈ Fin)
 
Theoremelfpw 9130 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 9131 A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝒫 𝐴 ∩ Fin) = 𝐴
 
Theoremf1opwfi 9132* 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 9133* 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 9134* 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 9135* 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 23205 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 9136* 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 35901. (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))
 
2.4.31  Finitely supported functions
 
Syntaxcfsupp 9137 Extend class definition to include the predicate to be a finitely supported function.
class finSupp
 
Definitiondf-fsupp 9138* 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 9139 The property of a function to be finitely supported is a relation. (Contributed by AV, 7-Jun-2019.)
Rel finSupp
 
Theoremrelprcnfsupp 9140 A proper class is never finitely supported. (Contributed by AV, 7-Jun-2019.)
𝐴 ∈ V → ¬ 𝐴 finSupp 𝑍)
 
Theoremisfsupp 9141 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 9142 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 9143 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 9144 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 9145 A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝐶) ∈ Fin)
 
Theoremfdmfisuppfi 9146 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 9147 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 9148* 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 9149 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 9150 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 9151 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 9152 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 9153 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 9154 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 9155 A constant function with value zero is finitely supported. (Contributed by AV, 30-Jun-2019.)
(𝜑𝐵𝑉)    &   (𝜑𝑍𝑊)       (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍)
 
Theoremfsuppun 9156 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 9157 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 9158 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 9159 The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
(𝑍𝑉 → ∅ finSupp 𝑍)
 
Theoremsnopfsupp 9160 A singleton containing an ordered pair is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩} finSupp 𝑍)
 
Theoremfunsnfsupp 9161 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 9162 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 9163 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 9164 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 9165 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 9166 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 9167* 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)
 
Theoremsniffsupp 9168* 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 )
 
Theoremfsuppcolem 9169 Lemma for fsuppco 9170. 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 9170 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 9171 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 9172 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 9172 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 9173* Lemma 1 for mapfien 9176. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
𝑆 = {𝑥 ∈ (𝐵m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺𝑍)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   (𝜑𝑍𝐵)       ((𝜑𝑓𝑆) → (𝐺 ∘ (𝑓𝐹)) finSupp 𝑊)
 
Theoremmapfienlem2 9174* Lemma 2 for mapfien 9176. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
𝑆 = {𝑥 ∈ (𝐵m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺𝑍)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   (𝜑𝑍𝐵)       ((𝜑𝑔𝑇) → ((𝐺𝑔) ∘ 𝐹) finSupp 𝑍)
 
Theoremmapfienlem3 9175* Lemma 3 for mapfien 9176. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
𝑆 = {𝑥 ∈ (𝐵m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺𝑍)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   (𝜑𝑍𝐵)       ((𝜑𝑔𝑇) → ((𝐺𝑔) ∘ 𝐹) ∈ 𝑆)
 
Theoremmapfien 9176* 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.) (Revised by AV, 28-Jul-2024.)
𝑆 = {𝑥 ∈ (𝐵m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺𝑍)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑓𝑆 ↦ (𝐺 ∘ (𝑓𝐹))):𝑆1-1-onto𝑇)
 
Theoremmapfien2 9177* 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)
 
2.4.32  Finite intersections
 
Syntaxcfi 9178 Extend class notation with the function whose value is the class of finite intersections of the elements of a given set.
class fi
 
Definitiondf-fi 9179* Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 9182). (Contributed by FL, 27-Apr-2008.)
fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = 𝑦})
 
Theoremfival 9180* The set of all the finite intersections of the elements of 𝐴. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
 
Theoremelfi 9181* Specific properties of an element of (fi‘𝐵). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
((𝐴𝑉𝐵𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
 
Theoremelfi2 9182* The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐵𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥))
 
Theoremelfir 9183 Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.)
((𝐵𝑉 ∧ (𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ (fi‘𝐵))
 
Theoremintrnfi 9184 Sufficient condition for the intersection of the range of a function to be in the set of finite intersections. (Contributed by Mario Carneiro, 30-Aug-2015.)
((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ∈ (fi‘𝐵))
 
Theoremiinfi 9185* An indexed intersection of elements of 𝐶 is an element of the finite intersections of 𝐶. (Contributed by Mario Carneiro, 30-Aug-2015.)
((𝐶𝑉 ∧ (∀𝑥𝐴 𝐵𝐶𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝑥𝐴 𝐵 ∈ (fi‘𝐶))
 
Theoreminelfi 9186 The intersection of two sets is a finite intersection. (Contributed by Thierry Arnoux, 6-Jan-2017.)
((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐵) ∈ (fi‘𝑋))
 
Theoremssfii 9187 Any element of a set 𝐴 is the intersection of a finite subset of 𝐴. (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
(𝐴𝑉𝐴 ⊆ (fi‘𝐴))
 
Theoremfi0 9188 The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
(fi‘∅) = ∅
 
Theoremfieq0 9189 A set is empty iff the class of all the finite intersections of that set is empty. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
 
Theoremfiin 9190 The elements of (fi‘𝐶) are closed under finite intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)
((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴𝐵) ∈ (fi‘𝐶))
 
Theoremdffi2 9191* The set of finite intersections is the smallest set that contains 𝐴 and is closed under pairwise intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 → (fi‘𝐴) = {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)})
 
Theoremfiss 9192 Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
((𝐵𝑉𝐴𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵))
 
Theoreminficl 9193* A set which is closed under pairwise intersection is closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
 
Theoremfipwuni 9194 The set of finite intersections of a set is contained in the powerset of the union of the elements of 𝐴. (Contributed by Mario Carneiro, 24-Nov-2013.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
(fi‘𝐴) ⊆ 𝒫 𝐴
 
Theoremfisn 9195 A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.)
(fi‘{𝐴}) = {𝐴}
 
Theoremfiuni 9196 The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 𝐴 = (fi‘𝐴))
 
Theoremfipwss 9197 If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.)
(𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋)
 
Theoremelfiun 9198* A finite intersection of elements taken from a union of collections. (Contributed by Jeff Hankins, 15-Nov-2009.) (Proof shortened by Mario Carneiro, 26-Nov-2013.)
((𝐵𝐷𝐶𝐾) → (𝐴 ∈ (fi‘(𝐵𝐶)) ↔ (𝐴 ∈ (fi‘𝐵) ∨ 𝐴 ∈ (fi‘𝐶) ∨ ∃𝑥 ∈ (fi‘𝐵)∃𝑦 ∈ (fi‘𝐶)𝐴 = (𝑥𝑦))))
 
Theoremdffi3 9199* The set of finite intersections can be "constructed" inductively by iterating binary intersection ω-many times. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑅 = (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)))       (𝐴𝑉 → (fi‘𝐴) = (rec(𝑅, 𝐴) “ ω))
 
Theoremfifo 9200* Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)
𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑦)       (𝐴𝑉𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
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