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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 8277 | . . . 4 ⊢ Rel finSupp | |
2 | 1 | brrelexi 5158 | . . 3 ⊢ (𝑅 finSupp 𝑍 → 𝑅 ∈ V) |
3 | 1 | brrelex2i 5159 | . . 3 ⊢ (𝑅 finSupp 𝑍 → 𝑍 ∈ V) |
4 | 2, 3 | jca 554 | . 2 ⊢ (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V)) |
5 | isfsupp 8279 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) | |
6 | 5 | biimpd 219 | . 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 384 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 Fun wfun 5882 (class class class)co 6650 supp csupp 7295 Fincfn 7955 finSupp cfsupp 8275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-fsupp 8276 |
This theorem is referenced by: fsuppimpd 8282 fsuppunfi 8295 fsuppunbi 8296 fsuppres 8300 fsuppco 8307 oemapvali 8581 mptnn0fsuppr 12799 gsumzres 18310 gsumzf1o 18313 |
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