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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 7134 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝑉) |
4 | fsuppmptdm.a | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
5 | fsuppmptdm.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
6 | 3, 4, 5 | fdmfifsupp 9413 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ↦ cmpt 5231 Fincfn 8984 finSupp cfsupp 9399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8185 df-1o 8505 df-en 8985 df-fin 8988 df-fsupp 9400 |
This theorem is referenced by: gsummptfidmadd 19958 gsummptfidmsplit 19963 gsummptfidmsplitres 19964 gsummptshft 19969 gsummptfidminv 19980 gsummptfidmsub 19983 gsumzunsnd 19989 gsummptf1o 19996 srgbinomlem3 20246 srgbinomlem4 20247 psrass1 22002 mamuass 22422 mamuvs1 22425 mamuvs2 22426 dmatmul 22519 mavmulass 22571 mdetrsca 22625 smadiadetlem3 22690 mat2pmatmul 22753 decpmatmul 22794 cpmadugsumlemB 22896 cpmadugsumlemC 22897 tsmsxplem1 24177 tsmsxplem2 24178 plypf1 26266 taylpfval 26421 lgseisenlem3 27436 lgseisenlem4 27437 gsummpt2d 33035 gsummptres 33038 gsummulgc2 33046 gsumvsca1 33215 gsumvsca2 33216 mdetpmtr1 33784 esumpfinval 34056 aacllem 49032 |
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