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Theorem xpcoidgend 13936
Description: If two classes are not disjoint, then the composition of their cross-product with itself is idempotent. (Contributed by RP, 24-Dec-2019.)
Hypothesis
Ref Expression
xpcoidgend.1 (𝜑 → (𝐴𝐵) ≠ ∅)
Assertion
Ref Expression
xpcoidgend (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))

Proof of Theorem xpcoidgend
StepHypRef Expression
1 incom 3949 . . 3 (𝐴𝐵) = (𝐵𝐴)
2 xpcoidgend.1 . . 3 (𝜑 → (𝐴𝐵) ≠ ∅)
31, 2syl5eqner 3008 . 2 (𝜑 → (𝐵𝐴) ≠ ∅)
43xpcogend 13935 1 (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wne 2933  cin 3715  c0 4059   × cxp 5265  ccom 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-co 5276
This theorem is referenced by:  xptrrel  13941  relexpxpnnidm  38516
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