Theorem xrnres3 39136

Description: Two ways to express restriction of range Cartesian product, see also xrnres 39134, xrnres2 39135. (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 6253	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = (◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴))
2		resco 6253	. . 3 ⊢ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴))
3	1, 2	ineq12i 4171	. 2 ⊢ (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴)) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴)))
4		df-xrn 39089	. . . 4 ⊢ (𝑅 ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
5	4	reseq1i 5976	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6		resindir 5997	. . 3 ⊢ (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
7	5, 6	eqtri 2788	. 2 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
8		df-xrn 39089	. 2 ⊢ ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴)) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴)))
9	3, 7, 8	3eqtr4i 2798	1 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3457   ∩ cin 3905   × cxp 5661  ^◡ccnv 5662   ↾ cres 5665   ∘ ccom 5667  1^st c1st 7990  2^nd c2nd 7991   ⋉ cxrn 38883

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-co 5672  df-res 5675  df-xrn 39089

This theorem is used by:  xrnres4  39137  xrnresex  39138