![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fsuppimp | Structured version Visualization version GIF version |
Description: Implications of a class being a finitely supported function (in relation to a given zero). (Contributed by AV, 26-May-2019.) |
Ref | Expression |
---|---|
fsuppimp | ⊢ (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relfsupp 9407 | . . 3 ⊢ Rel finSupp | |
2 | 1 | brrelex12i 5737 | . 2 ⊢ (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V)) |
3 | isfsupp 9409 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) | |
4 | 3 | biimpd 228 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) |
5 | 2, 4 | mpcom 38 | 1 ⊢ (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 Vcvv 3462 class class class wbr 5153 Fun wfun 6548 (class class class)co 7424 supp csupp 8174 Fincfn 8974 finSupp cfsupp 9405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-iota 6506 df-fun 6556 df-fv 6562 df-ov 7427 df-fsupp 9406 |
This theorem is referenced by: fsuppimpd 9413 fsuppfund 9414 fsuppunfi 9431 fsuppunbi 9432 fsuppres 9436 fsuppco 9445 oemapvali 9727 mptnn0fsuppr 14019 gsumzres 19907 gsumzf1o 19910 |
Copyright terms: Public domain | W3C validator |