Theorem xrneq2 38523

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 5805	. . 3 ⊢ (𝐴 = 𝐵 → (◡(2nd ↾ (V × V)) ∘ 𝐴) = (◡(2nd ↾ (V × V)) ∘ 𝐵))
2	1	ineq2d 4170	. 2 ⊢ (𝐴 = 𝐵 → ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)))
3		df-xrn 38504	. 2 ⊢ (𝐶 ⋉ 𝐴) = ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐴))
4		df-xrn 38504	. 2 ⊢ (𝐶 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐶) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
5	2, 3, 4	3eqtr4g 2794	1 ⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))

Syntax hints:   → wi 4   = wceq 1541  Vcvv 3438   ∩ cin 3898   × cxp 5620  ^◡ccnv 5621   ↾ cres 5624   ∘ ccom 5626  1^st c1st 7929  2^nd c2nd 7930   ⋉ cxrn 38314

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 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-in 3906  df-ss 3916  df-br 5097  df-opab 5159  df-co 5631  df-xrn 38504

This theorem is referenced by:  xrneq2i  38524  xrneq2d  38525  xrneq12  38526