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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 | fex 6745 | . . . 4 ⊢ ((𝐹:𝐷⟶𝑅 ∧ 𝐷 ∈ Fin) → 𝐹 ∈ V) | |
4 | 1, 2, 3 | syl2anc 581 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
5 | fdmfisuppfi.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
6 | suppimacnv 7570 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) | |
7 | 4, 5, 6 | syl2anc 581 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) |
8 | 2, 1 | fisuppfi 8552 | . 2 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) |
9 | 7, 8 | eqeltrd 2906 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 Vcvv 3414 ∖ cdif 3795 {csn 4397 ◡ccnv 5341 “ cima 5345 ⟶wf 6119 (class class class)co 6905 supp csupp 7559 Fincfn 8222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-supp 7560 df-er 8009 df-en 8223 df-fin 8226 |
This theorem is referenced by: fdmfifsupp 8554 fndmfisuppfi 8556 |
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