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Mirrors > Home > MPE Home > Th. List > Mathboxes > fresin2 | Structured version Visualization version GIF version |
Description: Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fresin2 | ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdm 5964 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | 1 | eqcomd 2616 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹) |
3 | 2 | ineq2d 3776 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ dom 𝐹)) |
4 | 3 | reseq2d 5317 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ (𝐶 ∩ dom 𝐹))) |
5 | frel 5963 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹) | |
6 | resindm 5364 | . . 3 ⊢ (Rel 𝐹 → (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹 ↾ 𝐶)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹 ↾ 𝐶)) |
8 | 4, 7 | eqtrd 2644 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∩ cin 3539 dom cdm 5038 ↾ cres 5040 Rel wrel 5043 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-dm 5048 df-res 5050 df-fun 5806 df-fn 5807 df-f 5808 |
This theorem is referenced by: (None) |
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