Theorem xrnres 39357

Description: Two ways to express restriction of range Cartesian product, see also xrnres2 39358, xrnres3 39359. (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 6251	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = (◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴))
2	1	ineq1i 4162	. 2 ⊢ (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴)) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
3		df-xrn 39312	. . . 4 ⊢ (𝑅 ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
4	3	reseq1i 5966	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
5		inres2 39179	. . 3 ⊢ (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6	4, 5	eqtr4i 2787	. 2 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
7		df-xrn 39312	. 2 ⊢ ((𝑅 ↾ 𝐴) ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ↾ 𝐴)) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
8	2, 6, 7	3eqtr4i 2794	1 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ 𝑆)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3451   ∩ cin 3898   × cxp 5649  ^◡ccnv 5650   ↾ cres 5653   ∘ ccom 5655  1^st c1st 7999  2^nd c2nd 8000   ⋉ cxrn 39106

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 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-co 5660  df-res 5663  df-xrn 39312

This theorem is used by:  dmxrncnvepres  39364