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Theorem resfifsupp 9367

Description: The restriction of a function to a finite set is finitely supported. (Contributed by AV, 12-Dec-2019.)

**Hypotheses**
Ref	Expression
resfifsupp.f	⊢ (𝜑 → Fun 𝐹)
resfifsupp.x	⊢ (𝜑 → 𝑋 ∈ Fin)
resfifsupp.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)

**Assertion**
Ref	Expression
resfifsupp	⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍)

**Proof of Theorem resfifsupp**
Step	Hyp	Ref	Expression
1		resindm 6023	. 2 ⊢ (𝐹 ↾ (𝑋 ∩ dom 𝐹)) = (𝐹 ↾ 𝑋)
2		resfifsupp.f	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
3	2	funfnd 6564	. . . 4 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
4		fnresin2 6658	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) Fn (𝑋 ∩ dom 𝐹))
5	3, 4	syl 18	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) Fn (𝑋 ∩ dom 𝐹))
6		resfifsupp.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin)
7		infi 9240	. . . 4 ⊢ (𝑋 ∈ Fin → (𝑋 ∩ dom 𝐹) ∈ Fin)
8	6, 7	syl 18	. . 3 ⊢ (𝜑 → (𝑋 ∩ dom 𝐹) ∈ Fin)
9		resfifsupp.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
10	5, 8, 9	fndmfifsupp 9348	. 2 ⊢ (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) finSupp 𝑍)
11	1, 10	eqbrtrrid 5141	1 ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∈ wcel 2145 ∩ cin 3898 class class class wbr 5103 dom cdm 5655 ↾ cres 5657 Fun wfun 6527 Fn wfn 6528 Fincfn 8952 finSupp cfsupp 9331

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pr 5398 ax-un 7736

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-supp 8159 df-1o 8455 df-en 8953 df-fin 8956 df-fsupp 9332

This theorem is used by: xrge0tsmsd 33513 ply1degltdimlem 34132

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