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Mirrors > Home > MPE Home > Th. List > fdmfifsupp | 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 |
---|---|
fdmfisuppfi.f | ⊢ (𝜑 → 𝐹:𝐷⟶𝑅) |
fdmfisuppfi.d | ⊢ (𝜑 → 𝐷 ∈ Fin) |
fdmfisuppfi.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
Ref | Expression |
---|---|
fdmfifsupp | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdmfisuppfi.f | . . 3 ⊢ (𝜑 → 𝐹:𝐷⟶𝑅) | |
2 | 1 | ffund 6511 | . 2 ⊢ (𝜑 → Fun 𝐹) |
3 | fdmfisuppfi.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ Fin) | |
4 | fdmfisuppfi.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
5 | 1, 3, 4 | fdmfisuppfi 8830 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
6 | 1 | ffnd 6508 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐷) |
7 | fnex 6971 | . . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐷 ∈ Fin) → 𝐹 ∈ V) | |
8 | 6, 3, 7 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
9 | isfsupp 8825 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 ↔ (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))) | |
10 | 8, 4, 9 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))) |
11 | 2, 5, 10 | mpbir2and 709 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 Vcvv 3492 class class class wbr 5057 Fun wfun 6342 Fn wfn 6343 ⟶wf 6344 (class class class)co 7145 supp csupp 7819 Fincfn 8497 finSupp cfsupp 8821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-supp 7820 df-er 8278 df-en 8498 df-fin 8501 df-fsupp 8822 |
This theorem is referenced by: fsuppmptdm 8832 fndmfifsupp 8834 gsumreidx 18966 gsummptfif1o 19017 psrmulcllem 20095 frlmfibas 20834 elfilspd 20875 tmdgsum 22631 tsmslem1 22664 tsmssubm 22678 tsmsres 22679 tsmsf1o 22680 tsmsmhm 22681 tsmsadd 22682 tsmsxplem1 22688 tsmsxplem2 22689 imasdsf1olem 22910 xrge0gsumle 23368 xrge0tsms 23369 rrxbasefi 23940 ehlbase 23945 jensenlem2 25492 jensen 25493 amgmlem 25494 amgm 25495 wilthlem2 25573 wilthlem3 25574 xrge0tsmsd 30619 gsumle 30652 linds2eq 30868 esumpfinvalf 31234 k0004ss2 40380 sge0tsms 42539 fsuppmptdmf 44357 linccl 44397 lcosn0 44403 islinindfis 44432 snlindsntor 44454 ldepspr 44456 zlmodzxzldeplem2 44484 amgmwlem 44831 amgmlemALT 44832 |
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