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Mathbox for Peter Mazsa |
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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 5836 | . . 3 ⊢ (𝑆 ∩ (𝑅 ↾ 𝐴)) = ((𝑆 ∩ 𝑅) ↾ 𝐴) | |
2 | 1 | ineqcomi 35665 | . 2 ⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑆 ∩ 𝑅) ↾ 𝐴) |
3 | incom 4128 | . . 3 ⊢ (𝑅 ∩ 𝑆) = (𝑆 ∩ 𝑅) | |
4 | 3 | reseq1i 5814 | . 2 ⊢ ((𝑅 ∩ 𝑆) ↾ 𝐴) = ((𝑆 ∩ 𝑅) ↾ 𝐴) |
5 | 2, 4 | eqtr4i 2824 | 1 ⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑅 ∩ 𝑆) ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∩ cin 3880 ↾ cres 5521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-in 3888 df-res 5531 |
This theorem is referenced by: xrnres 35810 |
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