Theorem xrnres2 39624

Description: Two ways to express restriction of range Cartesian product, see also xrnres 39623, xrnres3 39625. (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 6270	. . 3 ⊢ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴))
2	1	ineq2i 4117	. 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴)))
3		df-xrn 39578	. . . 4 ⊢ (𝑅 ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
4	3	reseq1i 5982	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
5		inres 6004	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6	4, 5	eqtr4i 2789	. 2 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
7		df-xrn 39578	. 2 ⊢ (𝑅 ⋉ (𝑆 ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴)))
8	2, 6, 7	3eqtr4i 2796	1 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3454   ∩ cin 4092   × cxp 5656  ^◡ccnv 5657   ↾ cres 5660   ∘ ccom 5662  1^st c1st 8033  2^nd c2nd 8034   ⋉ cxrn 39372

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-co 5667  df-res 5670  df-xrn 39578

This theorem is used by:  xrnresex  39627  dmxrncnvepres  39630  dfblockliftmap2  39659  br1cossxrnres  39736  disjxrnres5  40045