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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnresin | Structured version Visualization version GIF version |
Description: Restriction of a function with a subclass of its domain. (Contributed by Thierry Arnoux, 10-Oct-2017.) |
Ref | Expression |
---|---|
fnresin | ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐵) Fn (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresin1 6669 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵)) | |
2 | resindi 5991 | . . . 4 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵)) | |
3 | fnresdm 6663 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
4 | 3 | ineq1d 4206 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵)) = (𝐹 ∩ (𝐹 ↾ 𝐵))) |
5 | incom 4196 | . . . . . 6 ⊢ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ∩ (𝐹 ↾ 𝐵)) | |
6 | resss 6000 | . . . . . . 7 ⊢ (𝐹 ↾ 𝐵) ⊆ 𝐹 | |
7 | df-ss 3960 | . . . . . . 7 ⊢ ((𝐹 ↾ 𝐵) ⊆ 𝐹 ↔ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ↾ 𝐵)) | |
8 | 6, 7 | mpbi 229 | . . . . . 6 ⊢ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ↾ 𝐵) |
9 | 5, 8 | eqtr3i 2756 | . . . . 5 ⊢ (𝐹 ∩ (𝐹 ↾ 𝐵)) = (𝐹 ↾ 𝐵) |
10 | 4, 9 | eqtrdi 2782 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵)) = (𝐹 ↾ 𝐵)) |
11 | 2, 10 | eqtrid 2778 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ 𝐵)) |
12 | 11 | fneq1d 6636 | . 2 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵) ↔ (𝐹 ↾ 𝐵) Fn (𝐴 ∩ 𝐵))) |
13 | 1, 12 | mpbid 231 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐵) Fn (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∩ cin 3942 ⊆ wss 3943 ↾ cres 5671 Fn wfn 6532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-res 5681 df-fun 6539 df-fn 6540 |
This theorem is referenced by: fsuppcurry1 32457 fsuppcurry2 32458 signstres 34116 |
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