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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 6668 | . 2 ⊢ (𝜑 → Fun 𝐹) |
3 | fdmfisuppfi.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ Fin) | |
4 | fdmfisuppfi.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
5 | 1, 3, 4 | fdmfisuppfi 9248 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
6 | 1 | ffnd 6665 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐷) |
7 | fnex 7162 | . . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐷 ∈ Fin) → 𝐹 ∈ V) | |
8 | 6, 3, 7 | syl2anc 585 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
9 | isfsupp 9243 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 ↔ (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))) | |
10 | 8, 4, 9 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))) |
11 | 2, 5, 10 | mpbir2and 712 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 Vcvv 3444 class class class wbr 5104 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 (class class class)co 7350 supp csupp 8060 Fincfn 8817 finSupp cfsupp 9239 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7663 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-supp 8061 df-1o 8380 df-en 8818 df-fin 8821 df-fsupp 9240 |
This theorem is referenced by: fsuppmptdm 9250 fndmfifsupp 9252 gsumreidx 19629 gsummptfif1o 19680 frlmfibas 21097 elfilspd 21138 psrmulcllem 21284 tmdgsum 23374 tsmslem1 23408 tsmssubm 23422 tsmsres 23423 tsmsf1o 23424 tsmsmhm 23425 tsmsadd 23426 tsmsxplem1 23432 tsmsxplem2 23433 imasdsf1olem 23654 xrge0gsumle 24124 xrge0tsms 24125 rrxbasefi 24702 ehlbase 24707 jensenlem2 26265 jensen 26266 amgmlem 26267 amgm 26268 wilthlem2 26346 wilthlem3 26347 xrge0tsmsd 31700 gsumle 31733 linds2eq 31968 elrspunidl 31999 esumpfinvalf 32455 k0004ss2 42225 sge0tsms 44412 fsuppmptdmf 46248 linccl 46286 lcosn0 46292 islinindfis 46321 snlindsntor 46343 ldepspr 46345 zlmodzxzldeplem2 46373 amgmwlem 47040 amgmlemALT 47041 |
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