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| Mirrors > Home > MPE Home > Th. List > isfsupp | Structured version Visualization version GIF version | ||
| Description: The property of a class to be a finitely supported function (in relation to a given zero). (Contributed by AV, 23-May-2019.) |
| Ref | Expression |
|---|---|
| isfsupp | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funeq 6586 | . . . 4 ⊢ (𝑟 = 𝑅 → (Fun 𝑟 ↔ Fun 𝑅)) | |
| 2 | 1 | adantr 480 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (Fun 𝑟 ↔ Fun 𝑅)) |
| 3 | oveq12 7440 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (𝑟 supp 𝑧) = (𝑅 supp 𝑍)) | |
| 4 | 3 | eleq1d 2826 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((𝑟 supp 𝑧) ∈ Fin ↔ (𝑅 supp 𝑍) ∈ Fin)) |
| 5 | 2, 4 | anbi12d 632 | . 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin) ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) |
| 6 | df-fsupp 9402 | . 2 ⊢ finSupp = {〈𝑟, 𝑧〉 ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)} | |
| 7 | 5, 6 | brabga 5539 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-fsupp 9402 |
| This theorem is referenced by: isfsuppd 9406 funisfsupp 9407 fsuppimp 9408 fdmfifsupp 9415 fczfsuppd 9426 fsuppmptif 9439 fsuppco2 9443 fsuppcor 9444 mndpfsupp 18780 gsumzadd 19940 gsumpt 19980 gsum2dlem2 19989 gsum2d 19990 gsum2d2lem 19991 mhpmulcl 22153 rmfsupp2 33242 elrspunidl 33456 naddcnff 43375 rmfsupp 48289 scmfsupp 48291 mptcfsupp 48293 |
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