Theorem xrneq1i 38283

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

Hypothesis

Ref	Expression
xrneq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
xrneq1i	⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)

Proof of Theorem xrneq1i

Step	Hyp	Ref	Expression
1		xrneq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		xrneq1 38282	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)

Syntax hints:   = wceq 1537   ⋉ cxrn 38083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3439  df-in 3977  df-ss 3987  df-br 5170  df-opab 5232  df-co 5708  df-xrn 38276

This theorem is referenced by: (None)