Theorem xrneq2 38420

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 5797	. . 3 ⊢ (𝐴 = 𝐵 → (◡(2nd ↾ (V × V)) ∘ 𝐴) = (◡(2nd ↾ (V × V)) ∘ 𝐵))
2	1	ineq2d 4167	. 2 ⊢ (𝐴 = 𝐵 → ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)))
3		df-xrn 38407	. 2 ⊢ (𝐶 ⋉ 𝐴) = ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐴))
4		df-xrn 38407	. 2 ⊢ (𝐶 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
5	2, 3, 4	3eqtr4g 2791	1 ⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))

Syntax hints:   → wi 4   = wceq 1541  Vcvv 3436   ∩ cin 3896   × cxp 5612  ^◡ccnv 5613   ↾ cres 5616   ∘ ccom 5618  1^st c1st 7919  2^nd c2nd 7920   ⋉ cxrn 38222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-in 3904  df-ss 3914  df-br 5090  df-opab 5152  df-co 5623  df-xrn 38407

This theorem is referenced by:  xrneq2i  38421  xrneq2d  38422  xrneq12  38423