Theorem xrneq12 37764

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 37758	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))
2		xrneq2 37761	. 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 1533   ⋉ cxrn 37553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-br 5142  df-opab 5204  df-co 5678  df-xrn 37752

This theorem is referenced by:  xrneq12i  37765  xrneq12d  37766