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Theorem List for Metamath Proof Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremunifi 8801 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 8802* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 8801 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 8769). (Contributed by NM, 11-Mar-2006.)
((𝐴 ≺ ω ∧ ∀𝑥𝐴 𝑥 ≺ ω) → 𝐴 ≺ ω)
 
Theoreminfssuni 8803* 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 8804 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 8805 Images of finite sets are finite. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((Fun 𝐹𝑋 ∈ Fin) → (𝐹𝑋) ∈ Fin)
 
Theorempwfilem 8806* Lemma for pwfi 8807. (Contributed by NM, 26-Mar-2007.)
𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))       (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
 
Theorempwfi 8807 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 8808 Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴m 𝐵) ∈ Fin)
 
Theoremixpfi 8809* 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 8810* 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 8811* A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴 ∈ Fin → (𝑥𝐴𝐵) ∈ Fin)
 
Theoremabrexfi 8812* An image set from a finite set is finite. (Contributed by Mario Carneiro, 13-Feb-2014.)
(𝐴 ∈ Fin → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ Fin)
 
Theoremcnvimamptfin 8813* A preimage of a mapping with a finite domain under any class is finite. In contrast to fisuppfi 8829, the range of the mapping needs not to be known. (Contributed by AV, 21-Dec-2018.)
(𝜑𝑁 ∈ Fin)       (𝜑 → ((𝑝𝑁𝑋) “ 𝑌) ∈ Fin)
 
Theoremelfpw 8814 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 8815 A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝒫 𝐴 ∩ Fin) = 𝐴
 
Theoremf1opwfi 8816* 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 8817* 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 8818* 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 8819* 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 22581 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 8820* 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 34890. (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))
 
2.4.29  Finitely supported functions
 
Syntaxcfsupp 8821 Extend class definition to include the predicate to be a finitely supported function.
class finSupp
 
Definitiondf-fsupp 8822* 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 8823 The property of a function to be finitely supported is a relation. (Contributed by AV, 7-Jun-2019.)
Rel finSupp
 
Theoremrelprcnfsupp 8824 A proper class is never finitely supported. (Contributed by AV, 7-Jun-2019.)
𝐴 ∈ V → ¬ 𝐴 finSupp 𝑍)
 
Theoremisfsupp 8825 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 8826 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 8827 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 8828 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 8829 A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝐶) ∈ Fin)
 
Theoremfdmfisuppfi 8830 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 8831 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 8832* 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 8833 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 8834 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 8835 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 8836 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 8837 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 8838 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 8839 A constant function with value zero is finitely supported. (Contributed by AV, 30-Jun-2019.)
(𝜑𝐵𝑉)    &   (𝜑𝑍𝑊)       (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍)
 
Theoremfsuppun 8840 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 8841 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 8842 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 8843 The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
(𝑍𝑉 → ∅ finSupp 𝑍)
 
Theoremsnopfsupp 8844 A singleton containing an ordered pair is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩} finSupp 𝑍)
 
Theoremfunsnfsupp 8845 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 8846 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 8847 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 8848 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 8849 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 8850 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 8851* 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 8852 Lemma for fsuppco 8853. 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 8853 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 8854 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 8855 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 8855 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 8856* Lemma 1 for mapfien 8859. (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 8857* Lemma 2 for mapfien 8859. (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 8858* Lemma 3 for mapfien 8859. (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 8859* 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 8860* 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 8861* 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 )
 
2.4.30  Finite intersections
 
Syntaxcfi 8862 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 8863* 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 8866). (Contributed by FL, 27-Apr-2008.)
fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = 𝑦})
 
Theoremfival 8864* 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 8865* 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 8866* 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 8867 Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.)
((𝐵𝑉 ∧ (𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ (fi‘𝐵))
 
Theoremintrnfi 8868 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 8869* 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 8870 The intersection of two sets is a finite intersection. (Contributed by Thierry Arnoux, 6-Jan-2017.)
((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐵) ∈ (fi‘𝑋))
 
Theoremssfii 8871 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 8872 The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
(fi‘∅) = ∅
 
Theoremfieq0 8873 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 8874 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 8875* 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 8876 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 8877* 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 8878 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 8879 A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.)
(fi‘{𝐴}) = {𝐴}
 
Theoremfiuni 8880 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 8881 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 8882* 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 8883* 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 8884* 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))
 
2.4.31  Hall's marriage theorem
 
Theoremmarypha1lem 8885* Core induction for Philip Hall's marriage theorem. (Contributed by Stefan O'Rear, 19-Feb-2015.)
(𝐴 ∈ Fin → (𝑏 ∈ Fin → ∀𝑐 ∈ 𝒫 (𝐴 × 𝑏)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐𝑑) → ∃𝑒 ∈ 𝒫 𝑐𝑒:𝐴1-1→V)))
 
Theoremmarypha1 8886* (Philip) Hall's marriage theorem, sufficiency: a finite relation contains an injection if there is no subset of its domain which would be forced to violate the pigeonhole principle. (Contributed by Stefan O'Rear, 20-Feb-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐶 ⊆ (𝐴 × 𝐵))    &   ((𝜑𝑑𝐴) → 𝑑 ≼ (𝐶𝑑))       (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴1-1𝐵)
 
Theoremmarypha2lem1 8887* Lemma for marypha2 8891. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))       𝑇 ⊆ (𝐴 × ran 𝐹)
 
Theoremmarypha2lem2 8888* Lemma for marypha2 8891. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))       𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
 
Theoremmarypha2lem3 8889* Lemma for marypha2 8891. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))       ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥)))
 
Theoremmarypha2lem4 8890* Lemma for marypha2 8891. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))       ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = (𝐹𝑋))
 
Theoremmarypha2 8891* Version of marypha1 8886 using a functional family of sets instead of a relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴⟶Fin)    &   ((𝜑𝑑𝐴) → 𝑑 (𝐹𝑑))       (𝜑 → ∃𝑔(𝑔:𝐴1-1→V ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ (𝐹𝑥)))
 
2.4.32  Supremum and infimum
 
Syntaxcsup 8892 Extend class notation to include supremum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class sup(𝐴, 𝐵, 𝑅)
 
Syntaxcinf 8893 Extend class notation to include infimum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class inf(𝐴, 𝐵, 𝑅)
 
Definitiondf-sup 8894* Define the supremum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the supremum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals whose square is less than 2; in this case the supremum is defined as the square root of 2 per sqrtval 14584. See dfsup2 8896 for alternate definition not requiring dummy variables. (Contributed by NM, 22-May-1999.)
sup(𝐴, 𝐵, 𝑅) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
 
Definitiondf-inf 8895 Define the infimum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the infimum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
 
Theoremdfsup2 8896 Quantifier free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.)
sup(𝐵, 𝐴, 𝑅) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
 
Theoremsupeq1 8897 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
(𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
 
Theoremsupeq1d 8898 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐵 = 𝐶)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
 
Theoremsupeq1i 8899 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐵 = 𝐶       sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
 
Theoremsupeq2 8900 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
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