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| Mirrors > Home > MPE Home > Th. List > Mathboxes > xrnres2 | Structured version Visualization version GIF version | ||
| Description: Two ways to express restriction of range Cartesian product, see also xrnres 38403, xrnres3 38405. (Contributed by Peter Mazsa, 6-Sep-2021.) |
| Ref | Expression |
|---|---|
| xrnres2 | ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | resco 6270 | . . 3 ⊢ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴)) | |
| 2 | 1 | ineq2i 4217 | . 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴))) |
| 3 | df-xrn 38372 | . . . 4 ⊢ (𝑅 ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) | |
| 4 | 3 | reseq1i 5993 | . . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴) |
| 5 | inres 6015 | . . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴) | |
| 6 | 4, 5 | eqtr4i 2768 | . 2 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) |
| 7 | df-xrn 38372 | . 2 ⊢ (𝑅 ⋉ (𝑆 ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴))) | |
| 8 | 2, 6, 7 | 3eqtr4i 2775 | 1 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 Vcvv 3480 ∩ cin 3950 × cxp 5683 ◡ccnv 5684 ↾ cres 5687 ∘ ccom 5689 1st c1st 8012 2nd c2nd 8013 ⋉ cxrn 38181 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-co 5694 df-res 5697 df-xrn 38372 |
| This theorem is referenced by: xrnresex 38407 br1cossxrnres 38449 disjxrnres5 38748 |
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