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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 6578 | . . . 4 ⊢ (𝑟 = 𝑅 → (Fun 𝑟 ↔ Fun 𝑅)) | |
2 | 1 | adantr 479 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (Fun 𝑟 ↔ Fun 𝑅)) |
3 | oveq12 7432 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (𝑟 supp 𝑧) = (𝑅 supp 𝑍)) | |
4 | 3 | eleq1d 2810 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((𝑟 supp 𝑧) ∈ Fin ↔ (𝑅 supp 𝑍) ∈ Fin)) |
5 | 2, 4 | anbi12d 630 | . 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 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 Fun wfun 6547 (class class class)co 7423 supp csupp 8173 Fincfn 8973 finSupp cfsupp 9401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4325 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-rel 5688 df-cnv 5689 df-co 5690 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7426 df-fsupp 9402 |
This theorem is referenced by: isfsuppd 9406 funisfsupp 9407 fsuppimp 9408 fdmfifsupp 9414 fczfsuppd 9425 fsuppmptif 9438 fsuppco2 9442 fsuppcor 9443 gsumzadd 19915 gsumpt 19955 gsum2dlem2 19964 gsum2d 19965 gsum2d2lem 19966 mhpmulcl 22135 rmfsupp2 33080 elrspunidl 33280 naddcnff 42965 rmfsupp 47690 mndpfsupp 47692 scmfsupp 47694 mptcfsupp 47696 |
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