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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 8546 | . . . 4 ⊢ Rel finSupp | |
2 | 1 | brrelex1i 5393 | . . 3 ⊢ (𝑅 finSupp 𝑍 → 𝑅 ∈ V) |
3 | 1 | brrelex2i 5394 | . . 3 ⊢ (𝑅 finSupp 𝑍 → 𝑍 ∈ V) |
4 | 2, 3 | jca 509 | . 2 ⊢ (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V)) |
5 | isfsupp 8548 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) | |
6 | 5 | biimpd 221 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) |
7 | 4, 6 | mpcom 38 | 1 ⊢ (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 Vcvv 3414 class class class wbr 4873 Fun wfun 6117 (class class class)co 6905 supp csupp 7559 Fincfn 8222 finSupp cfsupp 8544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-iota 6086 df-fun 6125 df-fv 6131 df-ov 6908 df-fsupp 8545 |
This theorem is referenced by: fsuppimpd 8551 fsuppunfi 8564 fsuppunbi 8565 fsuppres 8569 fsuppco 8576 oemapvali 8858 mptnn0fsuppr 13093 gsumzres 18663 gsumzf1o 18666 |
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