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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 8819 | . . 3 ⊢ Rel finSupp | |
2 | 1 | brrelex12i 5571 | . 2 ⊢ (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V)) |
3 | isfsupp 8821 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) | |
4 | 3 | biimpd 232 | . 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 399 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 Fun wfun 6318 (class class class)co 7135 supp csupp 7813 Fincfn 8492 finSupp cfsupp 8817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-fsupp 8818 |
This theorem is referenced by: fsuppimpd 8824 fsuppunfi 8837 fsuppunbi 8838 fsuppres 8842 fsuppco 8849 oemapvali 9131 mptnn0fsuppr 13362 gsumzres 19022 gsumzf1o 19025 |
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