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Mirrors > Home > MPE Home > Th. List > fdmfisuppfi | Structured version Visualization version GIF version |
Description: The support of a function with a finite domain is always finite. (Contributed by AV, 27-Apr-2019.) |
Ref | Expression |
---|---|
fdmfisuppfi.f | ⊢ (𝜑 → 𝐹:𝐷⟶𝑅) |
fdmfisuppfi.d | ⊢ (𝜑 → 𝐷 ∈ Fin) |
fdmfisuppfi.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
Ref | Expression |
---|---|
fdmfisuppfi | ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdmfisuppfi.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶𝑅) | |
2 | fdmfisuppfi.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Fin) | |
3 | 1, 2 | fexd 7223 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
4 | fdmfisuppfi.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
5 | suppimacnv 8153 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) | |
6 | 3, 4, 5 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) |
7 | 2, 1 | fisuppfi 9365 | . 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 1542 ∈ wcel 2107 Vcvv 3475 ∖ cdif 3943 {csn 4626 ◡ccnv 5673 “ cima 5677 ⟶wf 6535 (class class class)co 7403 supp csupp 8140 Fincfn 8934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-supp 8141 df-1o 8460 df-en 8935 df-fin 8938 |
This theorem is referenced by: fdmfifsupp 9368 fndmfisuppfi 9370 |
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