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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 6463 | . 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 3933 ↾ cres 5550 Fn wfn 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-fun 6350 df-fn 6351 |
This theorem is referenced by: pfxccat1 14052 satfn 32499 xlimconst2 41992 |
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