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| Mirrors > Home > MPE Home > Th. List > fsuppfund | Structured version Visualization version GIF version | ||
| Description: A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.) | 
| Ref | Expression | 
|---|---|
| fsuppfund.1 | ⊢ (𝜑 → 𝐹 finSupp 𝑍) | 
| Ref | Expression | 
|---|---|
| fsuppfund | ⊢ (𝜑 → Fun 𝐹) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fsuppfund.1 | . 2 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
| 2 | fsuppimp 9409 | . . 3 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
| 3 | 2 | simpld 494 | . 2 ⊢ (𝐹 finSupp 𝑍 → Fun 𝐹) | 
| 4 | 1, 3 | syl 17 | 1 ⊢ (𝜑 → Fun 𝐹) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5142 Fun wfun 6554 (class class class)co 7432 supp csupp 8186 Fincfn 8986 finSupp cfsupp 9402 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-fsupp 9403 | 
| This theorem is referenced by: fsuppss 9424 | 
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