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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 6524 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | 1 | eqcomd 2829 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹) |
3 | 2 | ineq2d 4191 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ dom 𝐹)) |
4 | 3 | reseq2d 5855 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ (𝐶 ∩ dom 𝐹))) |
5 | frel 6521 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹) | |
6 | resindm 5902 | . . 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 2858 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3937 dom cdm 5557 ↾ cres 5559 Rel wrel 5562 ⟶wf 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-dm 5567 df-res 5569 df-fun 6359 df-fn 6360 df-f 6361 |
This theorem is referenced by: (None) |
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