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Theorem fdmfisuppfi 9165

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 7123	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
4		fdmfisuppfi.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
5		suppimacnv 8010	. . 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 9164	. 2 ⊢ (𝜑 → (^◡𝐹 “ (V ∖ {𝑍})) ∈ Fin)
8	6, 7	eqeltrd 2834	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2101 Vcvv 3434 ∖ cdif 3886 {csn 4564 ^◡ccnv 5590 “ cima 5594 ⟶wf 6443 (class class class)co 7295 supp csupp 7997 Fincfn 8753

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7608

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-supp 7998 df-1o 8317 df-en 8754 df-fin 8757

This theorem is referenced by: fdmfifsupp 9166 fndmfisuppfi 9168

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