Theorem xrnres3 38104

Description: Two ways to express restriction of range Cartesian product, see also xrnres 38102, xrnres2 38103. (Contributed by Peter Mazsa, 28-Mar-2020.)

Assertion

Ref	Expression
xrnres3	⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴))

Proof of Theorem xrnres3

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))

Syntax hints:   = wceq 1534  Vcvv 3462   ∩ cin 3946   × cxp 5682  ^◡ccnv 5683   ↾ cres 5686   ∘ ccom 5688  1^st c1st 8003  2^nd 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