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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 9363 | . . 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 2098 class class class wbr 5138 Fun wfun 6527 (class class class)co 7401 supp csupp 8140 Fincfn 8934 finSupp cfsupp 9356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7404 df-fsupp 9357 |
This theorem is referenced by: fsuppss 41524 |
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