Theorem xrneq2 38650

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 5815	. . 3 ⊢ (𝐴 = 𝐵 → (◡(2nd ↾ (V × V)) ∘ 𝐴) = (◡(2nd ↾ (V × V)) ∘ 𝐵))
2	1	ineq2d 4174	. 2 ⊢ (𝐴 = 𝐵 → ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)))
3		df-xrn 38631	. 2 ⊢ (𝐶 ⋉ 𝐴) = ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐴))
4		df-xrn 38631	. 2 ⊢ (𝐶 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
5	2, 3, 4	3eqtr4g 2797	1 ⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))

Syntax hints:   → wi 4   = wceq 1542  Vcvv 3442   ∩ cin 3902   × cxp 5630  ^◡ccnv 5631   ↾ cres 5634   ∘ ccom 5636  1^st c1st 7941  2^nd c2nd 7942   ⋉ cxrn 38425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-in 3910  df-ss 3920  df-br 5101  df-opab 5163  df-co 5641  df-xrn 38631

This theorem is referenced by:  xrneq2i  38651  xrneq2d  38652  xrneq12  38653