Theorem xrnres2 36456

Description: Two ways to express restriction of range Cartesian product, see also xrnres 36455, xrnres3 36457. (Contributed by Peter Mazsa, 6-Sep-2021.)

Assertion

Ref	Expression
xrnres2	⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))

Proof of Theorem xrnres2

Step	Hyp	Ref	Expression
1		resco 6143	. . 3 ⊢ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴))
2	1	ineq2i 4140	. 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴)))
3		df-xrn 36428	. . . 4 ⊢ (𝑅 ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
4	3	reseq1i 5876	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
5		inres 5898	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6	4, 5	eqtr4i 2769	. 2 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
7		df-xrn 36428	. 2 ⊢ (𝑅 ⋉ (𝑆 ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴)))
8	2, 6, 7	3eqtr4i 2776	1 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))

Syntax hints:   = wceq 1539  Vcvv 3422   ∩ cin 3882   × cxp 5578  ^◡ccnv 5579   ↾ cres 5582   ∘ ccom 5584  1^st c1st 7802  2^nd c2nd 7803   ⋉ cxrn 36259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-co 5589  df-res 5592  df-xrn 36428

This theorem is referenced by:  xrnresex  36459  br1cossxrnres  36493