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| Mirrors > Home > MPE Home > Th. List > fnssresd | Structured version Visualization version GIF version | ||
| Description: Restriction of a function to a subclass of its domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| Ref | Expression |
|---|---|
| fnssresd.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
| fnssresd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
| Ref | Expression |
|---|---|
| fnssresd | ⊢ (𝜑 → (𝐹 ↾ 𝐵) Fn 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fnssresd.1 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
| 2 | fnssresd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
| 3 | fnssres 6691 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) Fn 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ⊆ wss 3951 ↾ cres 5687 Fn wfn 6556 |
| 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 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: rescnvimafod 7093 fssrescdmd 7146 fpwwe2lem7 10677 pfxccat1 14740 mdetrsca 22609 2ndresdju 32659 fdifsupp 32694 fdifsuppconst 32698 ply1gsumz 33619 dimkerim 33678 rmulccn 33927 subfacp1lem3 35187 satfn 35360 eqresfnbd 42273 tfsconcatrev 43361 ofoafg 43367 xlimconst2 45850 dvnprodlem1 45961 fcoreslem4 47078 isubgredg 47852 |
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