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| Mirrors > Home > MPE Home > Th. List > funresd | Structured version Visualization version GIF version | ||
| Description: A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| Ref | Expression |
|---|---|
| funresd.1 | ⊢ (𝜑 → Fun 𝐹) |
| Ref | Expression |
|---|---|
| funresd | ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funresd.1 | . 2 ⊢ (𝜑 → Fun 𝐹) | |
| 2 | funres 6608 | . 2 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴)) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↾ cres 5687 Fun wfun 6555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-ss 3968 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-res 5697 df-fun 6563 |
| This theorem is referenced by: fnssresb 6690 respreima 7086 fssrescdmd 7146 frrlem11 8321 frrlem12 8322 frrlem15 9797 gsumzadd 19940 gsum2dlem2 19989 nogesgn1ores 27719 noinfres 27767 noinfbnd2lem1 27775 cyclnumvtx 29820 trlsegvdeglem2 30240 sspg 30747 ssps 30749 sspn 30755 fresf1o 32641 fsupprnfi 32701 gsumhashmul 33064 limsupresxr 45781 liminfresxr 45782 afvco2 47188 |
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