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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 6878 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝑉) |
4 | fsuppmptdm.a | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
5 | fsuppmptdm.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
6 | 3, 4, 5 | fdmfifsupp 8843 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ↦ cmpt 5146 Fincfn 8509 finSupp cfsupp 8833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-supp 7831 df-er 8289 df-en 8510 df-fin 8513 df-fsupp 8834 |
This theorem is referenced by: gsummptfidmadd 19045 gsummptfidmsplit 19050 gsummptfidmsplitres 19051 gsummptshft 19056 gsummptfidminv 19067 gsummptfidmsub 19070 gsumzunsnd 19076 gsummptf1o 19083 srgbinomlem3 19292 srgbinomlem4 19293 psrass1 20185 mamuass 21011 mamuvs1 21014 mamuvs2 21015 dmatmul 21106 mavmulass 21158 mdetrsca 21212 smadiadetlem3 21277 mat2pmatmul 21339 decpmatmul 21380 cpmadugsumlemB 21482 cpmadugsumlemC 21483 tsmsxplem1 22761 tsmsxplem2 22762 plypf1 24802 taylpfval 24953 lgseisenlem3 25953 lgseisenlem4 25954 gsummpt2d 30687 gsummptres 30690 gsumvsca1 30854 gsumvsca2 30855 mdetpmtr1 31088 esumpfinval 31334 aacllem 44922 |
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