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Theorem xrneq12 38364
Description: Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
Assertion
Ref Expression
xrneq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem xrneq12
StepHypRef Expression
1 xrneq1 38358 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 xrneq2 38361 . 2 (𝐶 = 𝐷 → (𝐵𝐶) = (𝐵𝐷))
31, 2sylan9eq 2794 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  cxrn 38160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-in 3969  df-ss 3979  df-br 5148  df-opab 5210  df-co 5697  df-xrn 38352
This theorem is referenced by:  xrneq12i  38365  xrneq12d  38366
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