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Theorem List for Metamath Proof Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnnunifi 8501 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 8502* Lemma for unbnn 8506. 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 8503* Lemma for unbnn 8506. 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 8504* Lemma for unbnn 8506. 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 8505* Lemma for unbnn 8506. 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 8506* 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 8855 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 8507* Version of unbnn 8506 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 8508 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 8509 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 8510 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 8511 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 8512 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
(ω ≼ 𝐴𝐴 ≠ ∅)

Theoremfin2inf 8513 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 8514* 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 8515* 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 8516 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 8517 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 8518 If a union is finite, the operands are finite. Converse of unfi 8517. (Contributed by FL, 3-Aug-2009.)
((𝐴𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))

Theoremunfi2 8519 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 8517 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 8513). (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)

Theoremdifinf 8520 An infinite set 𝐴 minus a finite set is infinite. (Contributed by FL, 3-Aug-2009.)
((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ¬ (𝐴𝐵) ∈ Fin)

Theoremxpfi 8521 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 8522 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 8523 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 8524* 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 8525 An unordered pair is finite. (Contributed by NM, 22-Aug-2008.)
{𝐴, 𝐵} ∈ Fin

Theoremtpfi 8526 An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.)
{𝐴, 𝐵, 𝐶} ∈ Fin

Theoremfiint 8527* 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 8528 A version of fnex 6755 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 8529 An onto function implies dominance of domain over range, for finite sets. Unlike fodom 9681 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 8530* Equivalence of an onto mapping and dominance for a nonempty finite set. Unlike fodomb 9685 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 8531 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 8532 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 8533 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 8534 The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)
(𝐴 ∈ Fin → dom 𝐴 ∈ Fin)

Theoremfundmfibi 8535 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 8536 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 8537 A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)
(( I ↾ 𝐴) ∈ Fin ↔ 𝐴 ∈ Fin)

Theoremcnvfi 8538 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ∈ Fin → 𝐴 ∈ Fin)

Theoremrnfi 8539 The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ∈ Fin → ran 𝐴 ∈ Fin)

Theoremf1dmvrnfibi 8540 A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 8541. (Contributed by AV, 10-Jan-2020.)
((𝐴𝑉𝐹:𝐴1-1𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

Theoremf1vrnfibi 8541 A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 8540. (Contributed by AV, 10-Jan-2020.)
((𝐹𝑉𝐹:𝐴1-1𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

Theoremfofi 8542 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 8543 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 8544* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 8545. 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 8545 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 8546* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 8545 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 8513). (Contributed by NM, 11-Mar-2006.)
((𝐴 ≺ ω ∧ ∀𝑥𝐴 𝑥 ≺ ω) → 𝐴 ≺ ω)

Theoreminfssuni 8547* 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 8548 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 8549 Images of finite sets are finite. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((Fun 𝐹𝑋 ∈ Fin) → (𝐹𝑋) ∈ Fin)

Theorempwfilem 8550* Lemma for pwfi 8551. (Contributed by NM, 26-Mar-2007.)
𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))       (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)

Theorempwfi 8551 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 8552 Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝑚 𝐵) ∈ Fin)

Theoremixpfi 8553* 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 8554* 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 8555* A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴 ∈ Fin → (𝑥𝐴𝐵) ∈ Fin)

Theoremabrexfi 8556* An image set from a finite set is finite. (Contributed by Mario Carneiro, 13-Feb-2014.)
(𝐴 ∈ Fin → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ Fin)

Theoremcnvimamptfin 8557* A preimage of a mapping with a finite domain under any class is finite. In contrast to fisuppfi 8573, the range of the mapping needs not to be known. (Contributed by AV, 21-Dec-2018.)
(𝜑𝑁 ∈ Fin)       (𝜑 → ((𝑝𝑁𝑋) “ 𝑌) ∈ Fin)

Theoremelfpw 8558 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 8559 A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝒫 𝐴 ∩ Fin) = 𝐴

Theoremf1opwfi 8560* 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 8561* 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 8562* 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 8563* 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 22268 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 8564* 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 34163. (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 8565 Extend class definition to include the predicate to be a finitely supported function.
class finSupp

Definitiondf-fsupp 8566* 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 8567 The property of a function to be finitely supported is a relation. (Contributed by AV, 7-Jun-2019.)
Rel finSupp

Theoremrelprcnfsupp 8568 A proper class is never finitely supported. (Contributed by AV, 7-Jun-2019.)
𝐴 ∈ V → ¬ 𝐴 finSupp 𝑍)

Theoremisfsupp 8569 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 8570 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 8571 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 8572 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 8573 A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝐶) ∈ Fin)

Theoremfdmfisuppfi 8574 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 8575 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 8576* 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 8577 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 8578 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 8579 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 8580 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 8581 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 8582 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 8583 A constant function with value zero is finitely supported. (Contributed by AV, 30-Jun-2019.)
(𝜑𝐵𝑉)    &   (𝜑𝑍𝑊)       (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍)

Theoremfsuppun 8584 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 8585 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 8586 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 8587 The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
(𝑍𝑉 → ∅ finSupp 𝑍)

Theoremsnopfsupp 8588 A singleton containing an ordered pair is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩} finSupp 𝑍)

Theoremfunsnfsupp 8589 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 8590 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 8591 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 8592 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 8593 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 8594 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 8595* 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 8596 Lemma for fsuppco 8597. 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 8597 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 8598 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 8599 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 8599 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 8600* Lemma 1 for mapfien 8603. (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)

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