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Mirrors > Home > MPE Home > Th. List > fndmfifsupp | Structured version Visualization version GIF version |
Description: A function with a finite domain is always finitely supported. (Contributed by AV, 25-May-2019.) |
Ref | Expression |
---|---|
fndmfisuppfi.f | ⊢ (𝜑 → 𝐹 Fn 𝐷) |
fndmfisuppfi.d | ⊢ (𝜑 → 𝐷 ∈ Fin) |
fndmfisuppfi.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
Ref | Expression |
---|---|
fndmfifsupp | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fndmfisuppfi.f | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐷) | |
2 | dffn3 6302 | . . 3 ⊢ (𝐹 Fn 𝐷 ↔ 𝐹:𝐷⟶ran 𝐹) | |
3 | 1, 2 | sylib 210 | . 2 ⊢ (𝜑 → 𝐹:𝐷⟶ran 𝐹) |
4 | fndmfisuppfi.d | . 2 ⊢ (𝜑 → 𝐷 ∈ Fin) | |
5 | fndmfisuppfi.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
6 | 3, 4, 5 | fdmfifsupp 8573 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 4886 ran crn 5356 Fn wfn 6130 ⟶wf 6131 Fincfn 8241 finSupp cfsupp 8563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-supp 7577 df-er 8026 df-en 8242 df-fin 8245 df-fsupp 8564 |
This theorem is referenced by: resfifsupp 8591 gsummptcl 18752 esumgsum 30705 gsumesum 30719 matunitlindflem1 34031 matunitlindflem2 34032 |
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