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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 6569 | . . . 4 ⊢ (𝑟 = 𝑅 → (Fun 𝑟 ↔ Fun 𝑅)) | |
2 | 1 | adantr 482 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (Fun 𝑟 ↔ Fun 𝑅)) |
3 | oveq12 7418 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (𝑟 supp 𝑧) = (𝑅 supp 𝑍)) | |
4 | 3 | eleq1d 2819 | . . 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 9362 | . 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)} | |
7 | 5, 6 | brabga 5535 | 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 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5149 Fun wfun 6538 (class class class)co 7409 supp csupp 8146 Fincfn 8939 finSupp cfsupp 9361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-rel 5684 df-cnv 5685 df-co 5686 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-fsupp 9362 |
This theorem is referenced by: isfsuppd 9366 funisfsupp 9367 fsuppimp 9368 fdmfifsupp 9373 fczfsuppd 9381 fsuppmptif 9394 fsuppco2 9398 fsuppcor 9399 gsumzadd 19790 gsumpt 19830 gsum2dlem2 19839 gsum2d 19840 gsum2d2lem 19841 mhpmulcl 21692 rmfsupp2 32387 elrspunidl 32546 naddcnff 42112 rmfsupp 47050 mndpfsupp 47052 scmfsupp 47054 mptcfsupp 47056 |
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