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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 6567 | . . . 4 ⊢ (𝑟 = 𝑅 → (Fun 𝑟 ↔ Fun 𝑅)) | |
2 | 1 | adantr 479 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (Fun 𝑟 ↔ Fun 𝑅)) |
3 | oveq12 7420 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (𝑟 supp 𝑧) = (𝑅 supp 𝑍)) | |
4 | 3 | eleq1d 2816 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((𝑟 supp 𝑧) ∈ Fin ↔ (𝑅 supp 𝑍) ∈ Fin)) |
5 | 2, 4 | anbi12d 629 | . 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin) ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) |
6 | df-fsupp 9364 | . 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)} | |
7 | 5, 6 | brabga 5533 | 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 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 class class class wbr 5147 Fun wfun 6536 (class class class)co 7411 supp csupp 8148 Fincfn 8941 finSupp cfsupp 9363 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-rel 5682 df-cnv 5683 df-co 5684 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7414 df-fsupp 9364 |
This theorem is referenced by: isfsuppd 9368 funisfsupp 9369 fsuppimp 9370 fdmfifsupp 9375 fczfsuppd 9383 fsuppmptif 9396 fsuppco2 9400 fsuppcor 9401 gsumzadd 19831 gsumpt 19871 gsum2dlem2 19880 gsum2d 19881 gsum2d2lem 19882 mhpmulcl 21911 rmfsupp2 32657 elrspunidl 32820 naddcnff 42414 rmfsupp 47138 mndpfsupp 47140 scmfsupp 47142 mptcfsupp 47144 |
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