Theorem xrneq12i 38377

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

Hypotheses

Ref	Expression
xrneq12i.1	⊢ 𝐴 = 𝐵
xrneq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
xrneq12i	⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)

Proof of Theorem xrneq12i

Step	Hyp	Ref	Expression
1		xrneq12i.1	. 2 ⊢ 𝐴 = 𝐵
2		xrneq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		xrneq12 38376	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)

Syntax hints:   = wceq 1540   ⋉ cxrn 38175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-in 3924  df-ss 3934  df-br 5111  df-opab 5173  df-co 5650  df-xrn 38360

This theorem is referenced by:  xrnres4  38398  xrnresex  38399