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Theorem fdmfisuppfi 8836

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		fex 6983	. . . 4 ⊢ ((𝐹:𝐷⟶𝑅 ∧ 𝐷 ∈ Fin) → 𝐹 ∈ V)
4	1, 2, 3	syl2anc 586	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
5		fdmfisuppfi.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
6		suppimacnv 7835	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) = (^◡𝐹 “ (V ∖ {𝑍})))
7	4, 5, 6	syl2anc 586	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = (^◡𝐹 “ (V ∖ {𝑍})))
8	2, 1	fisuppfi 8835	. 2 ⊢ (𝜑 → (^◡𝐹 “ (V ∖ {𝑍})) ∈ Fin)
9	7, 8	eqeltrd 2913	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 {csn 4560 ^◡ccnv 5548 “ cima 5552 ⟶wf 6345 (class class class)co 7150 supp csupp 7824 Fincfn 8503

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-supp 7825 df-er 8283 df-en 8504 df-fin 8507

This theorem is referenced by: fdmfifsupp 8837 fndmfisuppfi 8839

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