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Theorem fdmfisuppfi 9443

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 7264	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
4		fdmfisuppfi.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
5		suppimacnv 8215	. . 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 9441	. 2 ⊢ (𝜑 → (^◡𝐹 “ (V ∖ {𝑍})) ∈ Fin)
8	6, 7	eqeltrd 2844	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ∖ cdif 3973 {csn 4648 ^◡ccnv 5699 “ cima 5703 ⟶wf 6569 (class class class)co 7448 supp csupp 8201 Fincfn 9003

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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-supp 8202 df-1o 8522 df-en 9004 df-fin 9007

This theorem is referenced by: fdmfifsupp 9444 fndmfisuppfi 9446

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