Theorem xrneq12i 35796

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 35795	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))
4	1, 2, 3	mp2an 691	1 ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)

Syntax hints:   = wceq 1538   ⋉ cxrn 35612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-br 5031  df-opab 5093  df-co 5528  df-xrn 35783

This theorem is referenced by:  xrnres4  35813  xrnresex  35814