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Mirrors > Home > MPE Home > Th. List > Mathboxes > inres2 | Structured version Visualization version GIF version |
Description: Two ways of expressing the restriction of an intersection. (Contributed by Peter Mazsa, 5-Jun-2021.) |
Ref | Expression |
---|---|
inres2 | ⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑅 ∩ 𝑆) ↾ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inres 5871 | . . 3 ⊢ (𝑆 ∩ (𝑅 ↾ 𝐴)) = ((𝑆 ∩ 𝑅) ↾ 𝐴) | |
2 | 1 | ineqcomi 35520 | . 2 ⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑆 ∩ 𝑅) ↾ 𝐴) |
3 | incom 4178 | . . 3 ⊢ (𝑅 ∩ 𝑆) = (𝑆 ∩ 𝑅) | |
4 | 3 | reseq1i 5849 | . 2 ⊢ ((𝑅 ∩ 𝑆) ↾ 𝐴) = ((𝑆 ∩ 𝑅) ↾ 𝐴) |
5 | 2, 4 | eqtr4i 2847 | 1 ⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑅 ∩ 𝑆) ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∩ cin 3935 ↾ cres 5557 |
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 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3943 df-res 5567 |
This theorem is referenced by: xrnres 35665 |
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