P. 93 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9201-9300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pwfir 9201	If the power set of a set is finite, then the set itself is finite. (Contributed by BTernaryTau, 7-Sep-2024.)
⊢ (𝒫 𝐵 ∈ Fin → 𝐵 ∈ Fin)
Theorem	pwfilem 9202*	Lemma for pwfi 9203. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5303. (Revised by BTernaryTau, 7-Sep-2024.)
⊢ 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))    ⇒   ⊢ (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
Theorem	pwfi 9203	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.) Avoid ax-pow 5303. (Revised by BTernaryTau, 7-Sep-2024.)
⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
Theorem	xpfi 9204	The Cartesian product of two finite sets is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5303. (Revised by BTernaryTau, 10-Jan-2025.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)
Theorem	3xpfi 9205	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)
Theorem	domunfican 9206	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 ∧ 𝐵 ≈ 𝐴) ∧ ((𝐴 ∩ 𝑋) = ∅ ∧ (𝐵 ∩ 𝑌) = ∅)) → ((𝐴 ∪ 𝑋) ≼ (𝐵 ∪ 𝑌) ↔ 𝑋 ≼ 𝑌))
Theorem	infcntss 9207*	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(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
Theorem	prfi 9208	An unordered pair is finite. For a shorter proof using ax-un 7668, see prfiALT 9209. (Contributed by NM, 22-Aug-2008.) Avoid ax-11 2160, ax-un 7668. (Revised by BTernaryTau, 13-Jan-2025.)
⊢ {𝐴, 𝐵} ∈ Fin
Theorem	prfiALT 9209	Shorter proof of prfi 9208 using ax-un 7668. (Contributed by NM, 22-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝐴, 𝐵} ∈ Fin
Theorem	tpfi 9210	An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.)
⊢ {𝐴, 𝐵, 𝐶} ∈ Fin
Theorem	fiint 9211*	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.) Use a separate setvar for the right-hand side and avoid ax-pow 5303. (Revised by BTernaryTau, 14-Jan-2025.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ 𝑧 ∈ Fin) → ∩ 𝑧 ∈ 𝐴))
Theorem	fodomfir 9212*	There exists a mapping from a finite set onto any nonempty set that it dominates, proved without using the Axiom of Power Sets (unlike fodomr 9041). (Contributed by BTernaryTau, 23-Jun-2025.)
⊢ ((𝐴 ∈ Fin ∧ ∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)
Theorem	fodomfib 9213*	Equivalence of an onto mapping and dominance for a nonempty finite set. Unlike fodomb 10414 for arbitrary sets, this theorem does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5303. (Revised by BTernaryTau, 23-Jun-2025.)
⊢ (𝐴 ∈ Fin → ((𝐴 ≠ ∅ ∧ ∃𝑓 𝑓:𝐴–onto→𝐵) ↔ (∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴)))
Theorem	fodomfiOLD 9214	Obsolete version of fodomfi 9196 as of 20-Jun-2025. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)
Theorem	fodomfibOLD 9215*	Obsolete version of fodomfib 9213 as of 23-Jun-2025. (Contributed by NM, 23-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ Fin → ((𝐴 ≠ ∅ ∧ ∃𝑓 𝑓:𝐴–onto→𝐵) ↔ (∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴)))
Theorem	fofinf1o 9216	Any surjection from one finite set to another of equal size must be a bijection. (Contributed by Mario Carneiro, 19-Aug-2014.)
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴–1-1-onto→𝐵)
Theorem	rneqdmfinf1o 9217	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:𝐴–1-1-onto→𝐴)
Theorem	fidomdm 9218	Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ (𝐹 ∈ Fin → dom 𝐹 ≼ 𝐹)
Theorem	dmfi 9219	The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)
⊢ (𝐴 ∈ Fin → dom 𝐴 ∈ Fin)
Theorem	fundmfibi 9220	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))
Theorem	resfnfinfin 9221	The restriction of a function to a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ 𝐵) ∈ Fin)
Theorem	residfi 9222	A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)
⊢ (( I ↾ 𝐴) ∈ Fin ↔ 𝐴 ∈ Fin)
Theorem	cnvfiALT 9223	Shorter proof of cnvfi 9085 using ax-pow 5303. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin)
Theorem	rnfi 9224	The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
Theorem	f1dmvrnfibi 9225	A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 9226. (Contributed by AV, 10-Jan-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
Theorem	f1vrnfibi 9226	A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 9225. (Contributed by AV, 10-Jan-2020.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
Theorem	iunfi 9227*	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 9228. 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 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)
Theorem	unifi 9228	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)
Theorem	unifi2 9229*	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 9228 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 9188). (Contributed by NM, 11-Mar-2006.)
⊢ ((𝐴 ≺ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ ω) → ∪ 𝐴 ≺ ω)
Theorem	infssuni 9230*	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) → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)
Theorem	unirnffid 9231	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.)
⊢ (𝜑 → 𝐹:𝑇⟶Fin)    &   ⊢ (𝜑 → 𝑇 ∈ Fin)    ⇒   ⊢ (𝜑 → ∪ ran 𝐹 ∈ Fin)
Theorem	mapfi 9232	Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ↑m 𝐵) ∈ Fin)
Theorem	ixpfi 9233*	A Cartesian product of finitely many finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) → X𝑥 ∈ 𝐴 𝐵 ∈ Fin)
Theorem	ixpfi2 9234*	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.)
⊢ (𝜑 → 𝐶 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → 𝐵 ⊆ {𝐷})    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ∈ Fin)
Theorem	mptfi 9235*	A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin)
Theorem	abrexfi 9236*	An image set from a finite set is finite. (Contributed by Mario Carneiro, 13-Feb-2014.)
⊢ (𝐴 ∈ Fin → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ Fin)
Theorem	cnvimamptfin 9237*	A preimage of a mapping with a finite domain under any class is finite. In contrast to fisuppfi 9255, the range of the mapping needs not to be known. (Contributed by AV, 21-Dec-2018.)
⊢ (𝜑 → 𝑁 ∈ Fin)    ⇒   ⊢ (𝜑 → (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ∈ Fin)
Theorem	elfpw 9238	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) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ Fin))
Theorem	unifpw 9239	A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ ∪ (𝒫 𝐴 ∩ Fin) = 𝐴
Theorem	f1opwfi 9240*	A one-to-one mapping induces a one-to-one mapping on finite subsets. (Contributed by Mario Carneiro, 25-Jan-2015.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐹 “ 𝑏)):(𝒫 𝐴 ∩ Fin)–1-1-onto→(𝒫 𝐵 ∩ Fin))
Theorem	fissuni 9241*	A finite subset of a union is covered by finitely many elements. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)
Theorem	fipreima 9242*	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 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹 “ 𝑐) = 𝐴)
Theorem	finsschain 9243*	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 23958 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)) → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧)
Theorem	indexfi 9244*	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 37773. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑐 ∈ Fin (𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑))
2.4.35  Finitely supported functions
Syntax	cfsupp 9245	Extend class definition to include the predicate to be a finitely supported function.
class finSupp
Definition	df-fsupp 9246*	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)}
Theorem	relfsupp 9247	The property of a function to be finitely supported is a relation. (Contributed by AV, 7-Jun-2019.)
⊢ Rel finSupp
Theorem	relprcnfsupp 9248	A proper class is never finitely supported. (Contributed by AV, 7-Jun-2019.)
⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 finSupp 𝑍)
Theorem	isfsupp 9249	The property of a class to be a finitely supported function (in relation to a given zero). (Contributed by AV, 23-May-2019.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
Theorem	isfsuppd 9250	Deduction form of isfsupp 9249. (Contributed by SN, 29-Jul-2024.)
⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → Fun 𝑅)    &   ⊢ (𝜑 → (𝑅 supp 𝑍) ∈ Fin)    ⇒   ⊢ (𝜑 → 𝑅 finSupp 𝑍)
Theorem	funisfsupp 9251	The property of a function to be finitely supported (in relation to a given zero). (Contributed by AV, 23-May-2019.)
⊢ ((Fun 𝑅 ∧ 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (𝑅 supp 𝑍) ∈ Fin))
Theorem	fsuppimp 9252	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))
Theorem	fsuppimpd 9253	A finitely supported function is a function with a finite support. (Contributed by AV, 6-Jun-2019.)
⊢ (𝜑 → 𝐹 finSupp 𝑍)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
Theorem	fsuppfund 9254	A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.)
⊢ (𝜑 → 𝐹 finSupp 𝑍)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	fisuppfi 9255	A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (◡𝐹 “ 𝐶) ∈ Fin)
Theorem	fidmfisupp 9256	A function with a finite domain is finitely supported. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐹:𝐷⟶𝑅)    &   ⊢ (𝜑 → 𝐷 ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
Theorem	finnzfsuppd 9257*	If a function is zero outside of a finite set, it has finite support. (Contributed by Rohan Ridenour, 13-May-2024.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 Fn 𝐷)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝐴 ∨ (𝐹‘𝑥) = 𝑍))    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
Theorem	fdmfisuppfi 9258	The support of a function with a finite domain is always finite. (Contributed by AV, 27-Apr-2019.)
⊢ (𝜑 → 𝐹:𝐷⟶𝑅)    &   ⊢ (𝜑 → 𝐷 ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
Theorem	fdmfifsupp 9259	A function with a finite domain is always finitely supported. (Contributed by AV, 25-May-2019.)
⊢ (𝜑 → 𝐹:𝐷⟶𝑅)    &   ⊢ (𝜑 → 𝐷 ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
Theorem	fsuppmptdm 9260*	A mapping with a finite domain is finitely supported. (Contributed by AV, 7-Jun-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
Theorem	fndmfisuppfi 9261	The support of a function with a finite domain is always finite. (Contributed by AV, 25-May-2019.)
⊢ (𝜑 → 𝐹 Fn 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
Theorem	fndmfifsupp 9262	A function with a finite domain is always finitely supported. (Contributed by AV, 25-May-2019.)
⊢ (𝜑 → 𝐹 Fn 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
Theorem	suppeqfsuppbi 9263	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.)
⊢ (((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍 ↔ 𝐺 finSupp 𝑍)))
Theorem	suppssfifsupp 9264	If the support of a function is a subset of a finite set, the function is finitely supported. (Contributed by AV, 15-Jul-2019.)
⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → 𝐺 finSupp 𝑍)
Theorem	fsuppsssupp 9265	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.)
⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ (𝐹 finSupp 𝑍 ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))) → 𝐺 finSupp 𝑍)
Theorem	fsuppsssuppgd 9266	If the support of a function is a subset of a finite support, it is finite. Deduction associated with fsuppsssupp 9265. (Contributed by SN, 6-Mar-2025.)
⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑂)    &   ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))    ⇒   ⊢ (𝜑 → 𝐺 finSupp 𝑍)
Theorem	fsuppss 9267	A subset of a finitely supported function is a finitely supported function. (Contributed by SN, 8-Mar-2025.)
⊢ (𝜑 → 𝐹 ⊆ 𝐺)    &   ⊢ (𝜑 → 𝐺 finSupp 𝑍)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
Theorem	fsuppssov1 9268*	Formula building theorem for finite support: operator with left annihilator. Finite support version of suppssov1 8127. (Contributed by SN, 26-Apr-2025.)
⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌)    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) finSupp 𝑍)
Theorem	fsuppxpfi 9269	The cartesian product of two finitely supported functions is finite. (Contributed by AV, 17-Jul-2019.)
⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → ((𝐹 supp 𝑍) × (𝐺 supp 𝑍)) ∈ Fin)
Theorem	fczfsuppd 9270	A constant function with value zero is finitely supported. (Contributed by AV, 30-Jun-2019.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍)
Theorem	fsuppun 9271	The union of two finitely supported functions is finitely supported (but not necessarily a function!). (Contributed by AV, 3-Jun-2019.)
⊢ (𝜑 → 𝐹 finSupp 𝑍)    &   ⊢ (𝜑 → 𝐺 finSupp 𝑍)    ⇒   ⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin)
Theorem	fsuppunfi 9272	The union of the support of two finitely supported functions is finite. (Contributed by AV, 1-Jul-2019.)
⊢ (𝜑 → 𝐹 finSupp 𝑍)    &   ⊢ (𝜑 → 𝐺 finSupp 𝑍)    ⇒   ⊢ (𝜑 → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin)
Theorem	fsuppunbi 9273	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 (𝐹 ∪ 𝐺))    ⇒   ⊢ (𝜑 → ((𝐹 ∪ 𝐺) finSupp 𝑍 ↔ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍)))
Theorem	0fsupp 9274	The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
⊢ (𝑍 ∈ 𝑉 → ∅ finSupp 𝑍)
Theorem	snopfsupp 9275	A singleton containing an ordered pair is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {⟨𝑋, 𝑌⟩} finSupp 𝑍)
Theorem	funsnfsupp 9276	Finite support for a function extended by a singleton. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by AV, 19-Jul-2019.)
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹)) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) finSupp 𝑍 ↔ 𝐹 finSupp 𝑍))
Theorem	fsuppres 9277	The restriction of a finitely supported function is finitely supported. (Contributed by AV, 14-Jul-2019.)
⊢ (𝜑 → 𝐹 finSupp 𝑍)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍)
Theorem	fmptssfisupp 9278*	The restriction of a mapping function has finite support if that function has finite support. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) finSupp 𝑍)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐵) finSupp 𝑍)
Theorem	ressuppfi 9279	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 𝐹 ∖ 𝐵) ∈ Fin)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵))    &   ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
Theorem	resfsupp 9280	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 𝐹 ∖ 𝐵) ∈ Fin)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑊)    &   ⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵))    &   ⊢ (𝜑 → 𝐺 finSupp 𝑍)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
Theorem	resfifsupp 9281	The restriction of a function to a finite set is finitely supported. (Contributed by AV, 12-Dec-2019.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍)
Theorem	ffsuppbi 9282	Two ways of saying that a function with known codomain is finitely supported. (Contributed by AV, 8-Jul-2019.)
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 finSupp 𝑍 ↔ (◡𝐹 “ (𝑆 ∖ {𝑍})) ∈ Fin)))
Theorem	fsuppmptif 9283*	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.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑍)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍)
Theorem	sniffsupp 9284*	A function mapping all but one arguments to zero is finitely supported. (Contributed by AV, 8-Jul-2019.)
⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 0 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 ))    ⇒   ⊢ (𝜑 → 𝐹 finSupp 0 )
Theorem	fsuppcolem 9285	Lemma for fsuppco 9286. Formula building theorem for finite supports: rearranging the index set. (Contributed by Stefan O'Rear, 21-Mar-2015.)
⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin)    &   ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌)    ⇒   ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin)
Theorem	fsuppco 9286	The composition of a 1-1 function with a finitely supported function is finitely supported. (Contributed by AV, 28-May-2019.)
⊢ (𝜑 → 𝐹 finSupp 𝑍)    &   ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍)
Theorem	fsuppco2 9287	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 9288 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.)
⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑍)    &   ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍)
Theorem	fsuppcor 9288	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.)
⊢ (𝜑 → 0 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑍)    &   ⊢ (𝜑 → (𝐺‘𝑍) = 0 )    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 )
Theorem	mapfienlem1 9289*	Lemma 1 for mapfien 9292. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   ⊢ 𝑊 = (𝐺‘𝑍)    &   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊)
Theorem	mapfienlem2 9290*	Lemma 2 for mapfien 9292. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   ⊢ 𝑊 = (𝐺‘𝑍)    &   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) finSupp 𝑍)
Theorem	mapfienlem3 9291*	Lemma 3 for mapfien 9292. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   ⊢ 𝑊 = (𝐺‘𝑍)    &   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ 𝑆)
Theorem	mapfien 9292*	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 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   ⊢ 𝑊 = (𝐺‘𝑍)    &   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹))):𝑆–1-1-onto→𝑇)
Theorem	mapfien2 9293*	Equinumerousity relation for sets of finitely supported functions. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 0 }    &   ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   ⊢ (𝜑 → 𝐴 ≈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≈ 𝐷)    &   ⊢ (𝜑 → 0 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐷)    ⇒   ⊢ (𝜑 → 𝑆 ≈ 𝑇)
2.4.36  Finite intersections
Syntax	cfi 9294	Extend class notation with the function whose value is the class of finite intersections of the elements of a given set.
class fi
Definition	df-fi 9295*	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 9298). (Contributed by FL, 27-Apr-2008.)
⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
Theorem	fival 9296*	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.)
⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥})
Theorem	elfi 9297*	Specific properties of an element of (fi‘𝐵). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥))
Theorem	elfi2 9298*	The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥))
Theorem	elfir 9299	Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.)
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ∈ (fi‘𝐵))
Theorem	intrnfi 9300	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.)
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ ran 𝐹 ∈ (fi‘𝐵))