Theorem xrnres2 38396

Description: Two ways to express restriction of range Cartesian product, see also xrnres 38395, xrnres3 38397. (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 6226	. . 3 ⊢ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴))
2	1	ineq2i 4183	. 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴)))
3		df-xrn 38360	. . . 4 ⊢ (𝑅 ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
4	3	reseq1i 5949	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
5		inres 5971	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6	4, 5	eqtr4i 2756	. 2 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
7		df-xrn 38360	. 2 ⊢ (𝑅 ⋉ (𝑆 ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴)))
8	2, 6, 7	3eqtr4i 2763	1 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))

Syntax hints:   = wceq 1540  Vcvv 3450   ∩ cin 3916   × cxp 5639  ^◡ccnv 5640   ↾ cres 5643   ∘ ccom 5645  1^st c1st 7969  2^nd c2nd 7970   ⋉ cxrn 38175

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-co 5650  df-res 5653  df-xrn 38360

This theorem is referenced by:  xrnresex  38399  dmxrncnvepres  38402  br1cossxrnres  38446  disjxrnres5  38746