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Theorem fdmfisuppfi 9067

Description: The support of a function with a finite domain is always finite. (Contributed by AV, 27-Apr-2019.)

**Hypotheses**
Ref	Expression
fdmfisuppfi.f	⊢ (𝜑 → 𝐹:𝐷⟶𝑅)
fdmfisuppfi.d	⊢ (𝜑 → 𝐷 ∈ Fin)
fdmfisuppfi.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)

**Assertion**
Ref	Expression
fdmfisuppfi	⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

**Proof of Theorem fdmfisuppfi**
Step	Hyp	Ref	Expression
1		fdmfisuppfi.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶𝑅)
2		fdmfisuppfi.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Fin)
3	1, 2	fexd 7085	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
4		fdmfisuppfi.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
5		suppimacnv 7961	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) = (^◡𝐹 “ (V ∖ {𝑍})))
6	3, 4, 5	syl2anc 583	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = (^◡𝐹 “ (V ∖ {𝑍})))
7	2, 1	fisuppfi 9066	. 2 ⊢ (𝜑 → (^◡𝐹 “ (V ∖ {𝑍})) ∈ Fin)
8	6, 7	eqeltrd 2839	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ∖ cdif 3880 {csn 4558 ^◡ccnv 5579 “ cima 5583 ⟶wf 6414 (class class class)co 7255 supp csupp 7948 Fincfn 8691

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-supp 7949 df-1o 8267 df-en 8692 df-fin 8695

This theorem is referenced by: fdmfifsupp 9068 fndmfisuppfi 9070

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