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Mirrors > Home > MPE Home > Th. List > fresin | Structured version Visualization version GIF version |
Description: An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
fresin | ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4227 | . . 3 ⊢ (𝐴 ∩ 𝑋) ⊆ 𝐴 | |
2 | fssres 6756 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ∩ 𝑋) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵) | |
3 | 1, 2 | mpan2 687 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵) |
4 | resres 5993 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑋) = (𝐹 ↾ (𝐴 ∩ 𝑋)) | |
5 | ffn 6716 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
6 | fnresdm 6668 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝐴) = 𝐹) |
8 | 7 | reseq1d 5979 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 ↾ 𝐴) ↾ 𝑋) = (𝐹 ↾ 𝑋)) |
9 | 4, 8 | eqtr3id 2784 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐴 ∩ 𝑋)) = (𝐹 ↾ 𝑋)) |
10 | 9 | feq1d 6701 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵 ↔ (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵)) |
11 | 3, 10 | mpbid 231 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∩ cin 3946 ⊆ wss 3947 ↾ cres 5677 Fn wfn 6537 ⟶wf 6538 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-fun 6544 df-fn 6545 df-f 6546 |
This theorem is referenced by: o1res 15508 limcresi 25634 dvreslem 25658 dvres2lem 25659 noreson 27399 mbfresfi 36837 ofoafg 42406 limcresiooub 44656 limcresioolb 44657 limcleqr 44658 limclner 44665 mbfres2cn 44972 fouriersw 45245 sge0less 45406 sge0ssre 45411 smfres 45804 |
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