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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrnres3 | Structured version Visualization version GIF version |
Description: Two ways to express restriction of range Cartesian product, see also xrnres 38102, xrnres2 38103. (Contributed by Peter Mazsa, 28-Mar-2020.) |
Ref | Expression |
---|---|
xrnres3 | ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resco 6263 | . . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = (◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴)) | |
2 | resco 6263 | . . 3 ⊢ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴)) | |
3 | 1, 2 | ineq12i 4211 | . 2 ⊢ (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴)) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴))) |
4 | df-xrn 38071 | . . . 4 ⊢ (𝑅 ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) | |
5 | 4 | reseq1i 5987 | . . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴) |
6 | resindir 6008 | . . 3 ⊢ (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) | |
7 | 5, 6 | eqtri 2754 | . 2 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) |
8 | df-xrn 38071 | . 2 ⊢ ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴)) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴))) | |
9 | 3, 7, 8 | 3eqtr4i 2764 | 1 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 Vcvv 3462 ∩ cin 3946 × cxp 5682 ◡ccnv 5683 ↾ cres 5686 ∘ ccom 5688 1st c1st 8003 2nd c2nd 8004 ⋉ cxrn 37877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pr 5435 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5156 df-opab 5218 df-xp 5690 df-rel 5691 df-co 5693 df-res 5696 df-xrn 38071 |
This theorem is referenced by: xrnres4 38105 xrnresex 38106 |
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