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Mirrors > Home > MPE Home > Th. List > fsuppmptdm | Structured version Visualization version GIF version |
Description: A mapping with a finite domain is finitely supported. (Contributed by AV, 7-Jun-2019.) |
Ref | Expression |
---|---|
fsuppmptdm.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑌) |
fsuppmptdm.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsuppmptdm.y | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝑉) |
fsuppmptdm.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
Ref | Expression |
---|---|
fsuppmptdm | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppmptdm.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝑉) | |
2 | fsuppmptdm.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑌) | |
3 | 1, 2 | fmptd 6425 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝑉) |
4 | fsuppmptdm.a | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
5 | fsuppmptdm.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
6 | 3, 4, 5 | fdmfifsupp 8326 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 ↦ cmpt 4762 Fincfn 7997 finSupp cfsupp 8316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-supp 7341 df-er 7787 df-en 7998 df-fin 8001 df-fsupp 8317 |
This theorem is referenced by: gsummptfidmadd 18371 gsummptfidmsplit 18376 gsummptfidmsplitres 18377 gsummptshft 18382 gsummptfidminv 18393 gsummptfidmsub 18396 gsumzunsnd 18401 gsummptf1o 18408 srgbinomlem3 18588 srgbinomlem4 18589 psrass1 19453 mamuass 20256 mamuvs1 20259 mamuvs2 20260 dmatmul 20351 mavmulass 20403 mdetrsca 20457 smadiadetlem3 20522 mat2pmatmul 20584 decpmatmul 20625 cpmadugsumlemB 20727 cpmadugsumlemC 20728 tsmsxplem1 22003 tsmsxplem2 22004 plypf1 24013 taylpfval 24164 lgseisenlem3 25147 lgseisenlem4 25148 gsummpt2d 29909 gsumvsca1 29910 gsumvsca2 29911 gsummptres 29912 mdetpmtr1 30017 esumpfinval 30265 aacllem 42875 |
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