Theorem xrneq2 38403

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

Assertion

Ref	Expression
xrneq2	⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))

Proof of Theorem xrneq2

Step	Hyp	Ref	Expression
1		coeq2 5843	. . 3 ⊢ (𝐴 = 𝐵 → (◡(2nd ↾ (V × V)) ∘ 𝐴) = (◡(2nd ↾ (V × V)) ∘ 𝐵))
2	1	ineq2d 4200	. 2 ⊢ (𝐴 = 𝐵 → ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)))
3		df-xrn 38394	. 2 ⊢ (𝐶 ⋉ 𝐴) = ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐴))
4		df-xrn 38394	. 2 ⊢ (𝐶 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
5	2, 3, 4	3eqtr4g 2796	1 ⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))

Syntax hints:   → wi 4   = wceq 1540  Vcvv 3464   ∩ cin 3930   × cxp 5657  ^◡ccnv 5658   ↾ cres 5661   ∘ ccom 5663  1^st c1st 7991  2^nd c2nd 7992   ⋉ cxrn 38203

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-in 3938  df-ss 3948  df-br 5125  df-opab 5187  df-co 5668  df-xrn 38394

This theorem is referenced by:  xrneq2i  38404  xrneq2d  38405  xrneq12  38406