![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6587 | . . . 4 ⊢ (𝑟 = 𝑅 → (Fun 𝑟 ↔ Fun 𝑅)) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (Fun 𝑟 ↔ Fun 𝑅)) |
3 | oveq12 7439 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (𝑟 supp 𝑧) = (𝑅 supp 𝑍)) | |
4 | 3 | eleq1d 2823 | . . 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 9399 | . 2 ⊢ finSupp = {〈𝑟, 𝑧〉 ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)} | |
7 | 5, 6 | brabga 5543 | 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 1536 ∈ wcel 2105 class class class wbr 5147 Fun wfun 6556 (class class class)co 7430 supp csupp 8183 Fincfn 8983 finSupp cfsupp 9398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-rel 5695 df-cnv 5696 df-co 5697 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-fsupp 9399 |
This theorem is referenced by: isfsuppd 9403 funisfsupp 9404 fsuppimp 9405 fdmfifsupp 9412 fczfsuppd 9423 fsuppmptif 9436 fsuppco2 9440 fsuppcor 9441 mndpfsupp 18792 gsumzadd 19954 gsumpt 19994 gsum2dlem2 20003 gsum2d 20004 gsum2d2lem 20005 mhpmulcl 22170 rmfsupp2 33227 elrspunidl 33435 naddcnff 43351 rmfsupp 48217 scmfsupp 48219 mptcfsupp 48221 |
Copyright terms: Public domain | W3C validator |