Theorem xrneq1 38778

Description: Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)

Assertion

Ref	Expression
xrneq1	⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))

Proof of Theorem xrneq1

Step	Hyp	Ref	Expression
1		coeq2 5803	. . 3 ⊢ (𝐴 = 𝐵 → (◡(1st ↾ (V × V)) ∘ 𝐴) = (◡(1st ↾ (V × V)) ∘ 𝐵))
2	1	ineq1d 4151	. 2 ⊢ (𝐴 = 𝐵 → ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) = ((◡(1st ↾ (V × V)) ∘ 𝐵) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)))
3		df-xrn 38762	. 2 ⊢ (𝐴 ⋉ 𝐶) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶))
4		df-xrn 38762	. 2 ⊢ (𝐵 ⋉ 𝐶) = ((◡(1st ↾ (V × V)) ∘ 𝐵) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶))
5	2, 3, 4	3eqtr4g 2801	1 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))

Syntax hints:   → wi 4   = wceq 1548  Vcvv 3433   ∩ cin 3884   × cxp 5619  ^◡ccnv 5620   ↾ cres 5623   ∘ ccom 5625  1^st c1st 7933  2^nd c2nd 7934   ⋉ cxrn 38556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-in 3892  df-ss 3902  df-br 5076  df-opab 5138  df-co 5630  df-xrn 38762

This theorem is referenced by:  xrneq1i  38779  xrneq1d  38780  xrneq12  38784