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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 6692 | . 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 3963 ↾ cres 5691 Fn wfn 6558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-fun 6565 df-fn 6566 |
This theorem is referenced by: rescnvimafod 7093 fssrescdmd 7146 fpwwe2lem7 10675 pfxccat1 14737 mdetrsca 22625 2ndresdju 32666 fdifsupp 32700 fdifsuppconst 32704 ply1gsumz 33599 dimkerim 33655 rmulccn 33889 subfacp1lem3 35167 satfn 35340 eqresfnbd 42252 tfsconcatrev 43338 ofoafg 43344 xlimconst2 45791 dvnprodlem1 45902 fcoreslem4 47016 isubgredg 47790 |
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