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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 6873 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝑉) |
4 | fsuppmptdm.a | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
5 | fsuppmptdm.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
6 | 3, 4, 5 | fdmfifsupp 8837 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ↦ cmpt 5139 Fincfn 8503 finSupp cfsupp 8827 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-supp 7825 df-er 8283 df-en 8504 df-fin 8507 df-fsupp 8828 |
This theorem is referenced by: gsummptfidmadd 19039 gsummptfidmsplit 19044 gsummptfidmsplitres 19045 gsummptshft 19050 gsummptfidminv 19061 gsummptfidmsub 19064 gsumzunsnd 19070 gsummptf1o 19077 srgbinomlem3 19286 srgbinomlem4 19287 psrass1 20179 mamuass 21005 mamuvs1 21008 mamuvs2 21009 dmatmul 21100 mavmulass 21152 mdetrsca 21206 smadiadetlem3 21271 mat2pmatmul 21333 decpmatmul 21374 cpmadugsumlemB 21476 cpmadugsumlemC 21477 tsmsxplem1 22755 tsmsxplem2 22756 plypf1 24796 taylpfval 24947 lgseisenlem3 25947 lgseisenlem4 25948 gsummpt2d 30682 gsummptres 30685 gsumvsca1 30849 gsumvsca2 30850 mdetpmtr1 31083 esumpfinval 31329 aacllem 44895 |
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