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Theorem List for Metamath Proof Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremac6sfi 8201* A version of ac6s 9303 for finite sets. (Contributed by Jeff Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
(𝑦 = (𝑓𝑥) → (𝜑𝜓))       ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
 
Theoremfrfi 8202 A partial order is well-founded on a finite set. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
((𝑅 Po 𝐴𝐴 ∈ Fin) → 𝑅 Fr 𝐴)
 
Theoremfimax2g 8203* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
((𝑅 Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)
 
Theoremfimaxg 8204* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
((𝑅 Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑅𝑥))
 
Theoremfisupg 8205* Lemma showing existence and closure of supremum of a finite set. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑅 Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧)))
 
Theoremwofi 8206 A total order on a finite set is a well-order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝑅 We 𝐴)
 
Theoremordunifi 8207 The maximum of a finite collection of ordinals is in the set. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
 
Theoremnnunifi 8208 The union (supremum) of a finite set of finite ordinals is a finite ordinal. (Contributed by Stefan O'Rear, 5-Nov-2014.)
((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → 𝑆 ∈ ω)
 
Theoremunblem1 8209* Lemma for unbnn 8213. After removing the successor of an element from an unbounded set of natural numbers, the intersection of the result belongs to the original unbounded set. (Contributed by NM, 3-Dec-2003.)
(((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ∈ 𝐵)
 
Theoremunblem2 8210* Lemma for unbnn 8213. The value of the function 𝐹 belongs to the unbounded set of natural numbers 𝐴. (Contributed by NM, 3-Dec-2003.)
𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)       ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ 𝐴))
 
Theoremunblem3 8211* Lemma for unbnn 8213. The value of the function 𝐹 is less than its value at a successor. (Contributed by NM, 3-Dec-2003.)
𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)       ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ (𝐹‘suc 𝑧)))
 
Theoremunblem4 8212* Lemma for unbnn 8213. The function 𝐹 maps the set of natural numbers one-to-one to the set of unbounded natural numbers 𝐴. (Contributed by NM, 3-Dec-2003.)
𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)       ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐹:ω–1-1𝐴)
 
Theoremunbnn 8213* 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 8553 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 8214* Version of unbnn 8213 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 8215 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 8216 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 8217 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 8218 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 8219 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
(ω ≼ 𝐴𝐴 ≠ ∅)
 
Theoremfin2inf 8220 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 8221* 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 8222* 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 8223 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 8224 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 8225 If a union is finite, the operands are finite. Converse of unfi 8224. (Contributed by FL, 3-Aug-2009.)
((𝐴𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))
 
Theoremunfi2 8226 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 8224 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 8220). (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
 
Theoremdifinf 8227 An infinite set 𝐴 minus a finite set is infinite. (Contributed by FL, 3-Aug-2009.)
((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ¬ (𝐴𝐵) ∈ Fin)
 
Theoremxpfi 8228 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 8229 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 8230 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 8231* 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 8232 An unordered pair is finite. (Contributed by NM, 22-Aug-2008.)
{𝐴, 𝐵} ∈ Fin
 
Theoremtpfi 8233 An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.)
{𝐴, 𝐵, 𝐶} ∈ Fin
 
Theoremfiint 8234* 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 8235 A version of fnex 6478 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 8236 An onto function implies dominance of domain over range, for finite sets. Unlike fodom 9341 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 8237* Equivalence of an onto mapping and dominance for a nonempty finite set. Unlike fodomb 9345 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 8238 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 8239 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 8240 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 8241 The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)
(𝐴 ∈ Fin → dom 𝐴 ∈ Fin)
 
Theoremfundmfibi 8242 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))
 
Theoremresfnfinfin 8243 The restriction of a function by a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹𝐵) ∈ Fin)
 
Theoremresidfi 8244 A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)
(( I ↾ 𝐴) ∈ Fin ↔ 𝐴 ∈ Fin)
 
Theoremcnvfi 8245 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ∈ Fin → 𝐴 ∈ Fin)
 
Theoremrnfi 8246 The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
 
Theoremf1dmvrnfibi 8247 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 8248 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 8249 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 8250 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 8251* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 8252. 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 8252 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 8253* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 8252 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 8220). (Contributed by NM, 11-Mar-2006.)
((𝐴 ≺ ω ∧ ∀𝑥𝐴 𝑥 ≺ ω) → 𝐴 ≺ ω)
 
Theoreminfssuni 8254* 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 8255 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 8256 Images of finite sets are finite. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((Fun 𝐹𝑋 ∈ Fin) → (𝐹𝑋) ∈ Fin)
 
Theorempwfilem 8257* Lemma for pwfi 8258. (Contributed by NM, 26-Mar-2007.)
𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))       (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
 
Theorempwfi 8258 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 8259 Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝑚 𝐵) ∈ Fin)
 
Theoremixpfi 8260* 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 8261* 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 8262* A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴 ∈ Fin → (𝑥𝐴𝐵) ∈ Fin)
 
Theoremabrexfi 8263* An image set from a finite set is finite. (Contributed by Mario Carneiro, 13-Feb-2014.)
(𝐴 ∈ Fin → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ Fin)
 
Theoremcnvimamptfin 8264* A preimage of a mapping with a finite domain under any class is finite. In contrast to fisuppfi 8280, the range of the mapping needs not to be known. (Contributed by AV, 21-Dec-2018.)
(𝜑𝑁 ∈ Fin)       (𝜑 → ((𝑝𝑁𝑋) “ 𝑌) ∈ Fin)
 
Theoremelfpw 8265 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 8266 A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝒫 𝐴 ∩ Fin) = 𝐴
 
Theoremf1opwfi 8267* 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 8268* 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 8269* 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 8270* 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 21843 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 8271* 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 33509. (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 8272 Extend class definition to include the predicate to be a finitely supported function.
class finSupp
 
Definitiondf-fsupp 8273* 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 8274 The property of a function to be finitely supported is a relation. (Contributed by AV, 7-Jun-2019.)
Rel finSupp
 
Theoremrelprcnfsupp 8275 A proper class is never finitely supported. (Contributed by AV, 7-Jun-2019.)
𝐴 ∈ V → ¬ 𝐴 finSupp 𝑍)
 
Theoremisfsupp 8276 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 8277 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 8278 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 8279 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 8280 A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝐶) ∈ Fin)
 
Theoremfdmfisuppfi 8281 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 8282 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 8283* 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 8284 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 8285 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 8286 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 8287 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 8288 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 8289 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 8290 A constant function with value zero is finitely supported. (Contributed by AV, 30-Jun-2019.)
(𝜑𝐵𝑉)    &   (𝜑𝑍𝑊)       (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍)
 
Theoremfsuppun 8291 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 8292 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 8293 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 8294 The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
(𝑍𝑉 → ∅ finSupp 𝑍)
 
Theoremsnopfsupp 8295 A singleton containing an ordered pair is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩} finSupp 𝑍)
 
Theoremfunsnfsupp 8296 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 8297 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 8298 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 8299 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 8300 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)
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