Theorem xrnres 38676

Description: Two ways to express restriction of range Cartesian product, see also xrnres2 38677, xrnres3 38678. (Contributed by Peter Mazsa, 5-Jun-2021.)

Assertion

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

Proof of Theorem xrnres

Step	Hyp	Ref	Expression
1		resco 6216	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = (◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴))
2	1	ineq1i 4170	. 2 ⊢ (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴)) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
3		df-xrn 38631	. . . 4 ⊢ (𝑅 ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
4	3	reseq1i 5942	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
5		inres2 38498	. . 3 ⊢ (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6	4, 5	eqtr4i 2763	. 2 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
7		df-xrn 38631	. 2 ⊢ ((𝑅 ↾ 𝐴) ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴)) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
8	2, 6, 7	3eqtr4i 2770	1 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ 𝑆)

Syntax hints:   = wceq 1542  Vcvv 3442   ∩ cin 3902   × cxp 5630  ^◡ccnv 5631   ↾ cres 5634   ∘ ccom 5636  1^st c1st 7941  2^nd c2nd 7942   ⋉ cxrn 38425

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-co 5641  df-res 5644  df-xrn 38631

This theorem is referenced by:  dmxrncnvepres  38683