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| Mirrors > Home > MPE Home > Th. List > suppssfifsupp | Structured version Visualization version GIF version | ||
| Description: If the support of a function is a subset of a finite set, the function is finitely supported. (Contributed by AV, 15-Jul-2019.) |
| Ref | Expression |
|---|---|
| suppssfifsupp | ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → 𝐺 finSupp 𝑍) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssfi 9213 | . . 3 ⊢ ((𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹) → (𝐺 supp 𝑍) ∈ Fin) | |
| 2 | 1 | adantl 481 | . 2 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → (𝐺 supp 𝑍) ∈ Fin) |
| 3 | 3ancoma 1098 | . . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ↔ (Fun 𝐺 ∧ 𝐺 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) | |
| 4 | 3 | biimpi 216 | . . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) → (Fun 𝐺 ∧ 𝐺 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) |
| 5 | 4 | adantr 480 | . . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → (Fun 𝐺 ∧ 𝐺 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) |
| 6 | funisfsupp 9407 | . . 3 ⊢ ((Fun 𝐺 ∧ 𝐺 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)) | |
| 7 | 5, 6 | syl 17 | . 2 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)) |
| 8 | 2, 7 | mpbird 257 | 1 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → 𝐺 finSupp 𝑍) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 Fun wfun 6555 (class class class)co 7431 supp csupp 8185 Fincfn 8985 finSupp cfsupp 9401 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-1o 8506 df-en 8986 df-fin 8989 df-fsupp 9402 |
| This theorem is referenced by: fsuppsssupp 9421 fsuppsssuppgd 9422 fsfnn0gsumfsffz 20001 mptscmfsupp0 20925 uvcff 21811 uvcresum 21813 frlmup1 21818 psrass1lem 21952 psrlidm 21982 psrridm 21983 psrass1 21984 psrass23l 21987 psrcom 21988 psrass23 21989 mvrcl 22012 mplsubrglem 22024 mplsubrg 22025 mplmon 22053 mplmonmul 22054 mplcoe1 22055 mplcoe5 22058 mplbas2 22060 psrbagev1 22101 evlslem2 22103 evlslem3 22104 evlslem6 22105 psropprmul 22239 coe1mul2 22272 evls1fpws 22373 plypf1 26251 tayl0 26403 fsuppcurry1 32736 fsuppcurry2 32737 gsummptres2 33056 elrgspnlem2 33247 elrgspnlem3 33248 ply1degltdimlem 33673 fedgmullem1 33680 fedgmullem2 33681 evls1fldgencl 33720 lincresunit2 48395 |
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