Theorem xrnres 38924

Description: Two ways to express restriction of range Cartesian product, see also xrnres2 38925, xrnres3 38926. (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 6237	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = (◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴))
2	1	ineq1i 4168	. 2 ⊢ (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴)) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
3		df-xrn 38879	. . . 4 ⊢ (𝑅 ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
4	3	reseq1i 5961	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
5		inres2 38746	. . 3 ⊢ (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6	4, 5	eqtr4i 2788	. 2 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
7		df-xrn 38879	. 2 ⊢ ((𝑅 ↾ 𝐴) ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴)) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
8	2, 6, 7	3eqtr4i 2795	1 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ 𝑆)

Syntax hints:   = wceq 1560  Vcvv 3454   ∩ cin 3903   × cxp 5645  ^◡ccnv 5646   ↾ cres 5649   ∘ ccom 5651  1^st c1st 7968  2^nd c2nd 7969   ⋉ cxrn 38673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-co 5656  df-res 5659  df-xrn 38879

This theorem is referenced by:  dmxrncnvepres  38931