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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 6694 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵)) | |
2 | resindi 6016 | . . . 4 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵)) | |
3 | fnresdm 6688 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
4 | 3 | ineq1d 4227 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵)) = (𝐹 ∩ (𝐹 ↾ 𝐵))) |
5 | incom 4217 | . . . . . 6 ⊢ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ∩ (𝐹 ↾ 𝐵)) | |
6 | resss 6022 | . . . . . . 7 ⊢ (𝐹 ↾ 𝐵) ⊆ 𝐹 | |
7 | dfss2 3981 | . . . . . . 7 ⊢ ((𝐹 ↾ 𝐵) ⊆ 𝐹 ↔ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ↾ 𝐵)) | |
8 | 6, 7 | mpbi 230 | . . . . . 6 ⊢ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ↾ 𝐵) |
9 | 5, 8 | eqtr3i 2765 | . . . . 5 ⊢ (𝐹 ∩ (𝐹 ↾ 𝐵)) = (𝐹 ↾ 𝐵) |
10 | 4, 9 | eqtrdi 2791 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵)) = (𝐹 ↾ 𝐵)) |
11 | 2, 10 | eqtrid 2787 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ 𝐵)) |
12 | 11 | fneq1d 6662 | . 2 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵) ↔ (𝐹 ↾ 𝐵) Fn (𝐴 ∩ 𝐵))) |
13 | 1, 12 | mpbid 232 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐵) Fn (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3962 ⊆ 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: fsuppcurry1 32743 fsuppcurry2 32744 signstres 34569 |
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