resfifsupp - Metamath Proof Explorer

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Theorem resfifsupp 9382

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 6019	. 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 9254	. . . 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 9363	. 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 5651 ↾ cres 5653 Fun wfun 6531 Fn wfn 6532 Fincfn 8966 finSupp cfsupp 9346

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 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7749

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 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 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-supp 8171 df-1o 8469 df-en 8967 df-fin 8970 df-fsupp 9347

This theorem is used by: xrge0tsmsd 33627 ply1degltdimlem 34247

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