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Theorem fdmfisuppfi 9306

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 7196	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
4		fdmfisuppfi.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
5		suppimacnv 8138	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) = (^◡𝐹 “ (V ∖ {𝑍})))
6	3, 4, 5	syl2anc 592	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = (^◡𝐹 “ (V ∖ {𝑍})))
7	2, 1	fisuppfi 9303	. 2 ⊢ (𝜑 → (^◡𝐹 “ (V ∖ {𝑍})) ∈ Fin)
8	6, 7	eqeltrd 2852	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1550 ∈ wcel 2132 Vcvv 3444 ∖ cdif 3892 {csn 4572 ^◡ccnv 5635 “ cima 5639 ⟶wf 6502 (class class class)co 7381 supp csupp 8124 Fincfn 8912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pr 5380 ax-un 7703

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-ral 3067 df-rex 3077 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-iun 4941 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-ov 7384 df-oprab 7385 df-mpo 7386 df-om 7832 df-supp 8125 df-1o 8421 df-en 8913 df-fin 8916

This theorem is referenced by: fdmfifsupp 9307 fndmfisuppfi 9309

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