Theorem xrneq1 38370

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 5825	. . 3 ⊢ (𝐴 = 𝐵 → (◡(1st ↾ (V × V)) ∘ 𝐴) = (◡(1st ↾ (V × V)) ∘ 𝐵))
2	1	ineq1d 4185	. 2 ⊢ (𝐴 = 𝐵 → ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) = ((◡(1st ↾ (V × V)) ∘ 𝐵) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)))
3		df-xrn 38360	. 2 ⊢ (𝐴 ⋉ 𝐶) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶))
4		df-xrn 38360	. 2 ⊢ (𝐵 ⋉ 𝐶) = ((◡(1st ↾ (V × V)) ∘ 𝐵) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶))
5	2, 3, 4	3eqtr4g 2790	1 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))

Syntax hints:   → wi 4   = wceq 1540  Vcvv 3450   ∩ cin 3916   × cxp 5639  ^◡ccnv 5640   ↾ cres 5643   ∘ ccom 5645  1^st c1st 7969  2^nd c2nd 7970   ⋉ cxrn 38175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-in 3924  df-ss 3934  df-br 5111  df-opab 5173  df-co 5650  df-xrn 38360

This theorem is referenced by:  xrneq1i  38371  xrneq1d  38372  xrneq12  38376