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Theorem resfifsupp 9358

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 6031	. 2 ⊢ (𝐹 ↾ (𝑋 ∩ dom 𝐹)) = (𝐹 ↾ 𝑋)
2		resfifsupp.f	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
3	2	funfnd 6569	. . . 4 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
4		fnresin2 6663	. . . 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 9231	. . . 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 9339	. 2 ⊢ (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) finSupp 𝑍)
11	1, 10	eqbrtrrid 5148	1 ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2143 ∩ cin 3905 class class class wbr 5110 dom cdm 5663 ↾ cres 5665 Fun wfun 6532 Fn wfn 6533 Fincfn 8944 finSupp cfsupp 9322

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-supp 8158 df-1o 8454 df-en 8945 df-fin 8948 df-fsupp 9323

This theorem is referenced by: xrge0tsmsd 33374 ply1degltdimlem 33993

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