Theorem xrneq12i 36441

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 36440	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))
4	1, 2, 3	mp2an 688	1 ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)

Syntax hints:   = wceq 1539   ⋉ cxrn 36259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-br 5071  df-opab 5133  df-co 5589  df-xrn 36428

This theorem is referenced by:  xrnres4  36458  xrnresex  36459