Theorem xrneq12 38414

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

Assertion

Ref	Expression
xrneq12	⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))

Proof of Theorem xrneq12

Step	Hyp	Ref	Expression
1		xrneq1 38408	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))
2		xrneq2 38411	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))
3	1, 2	sylan9eq 2786	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ⋉ cxrn 38213

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 3909  df-ss 3919  df-br 5092  df-opab 5154  df-co 5625  df-xrn 38398

This theorem is referenced by:  xrneq12i  38415  xrneq12d  38416