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Theorem xpcoidgend 14054
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 4001 . . 3 (𝐴𝐵) = (𝐵𝐴)
2 xpcoidgend.1 . . 3 (𝜑 → (𝐴𝐵) ≠ ∅)
31, 2syl5eqner 3044 . 2 (𝜑 → (𝐵𝐴) ≠ ∅)
43xpcogend 14053 1 (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wne 2969  cin 3766  c0 4113   × cxp 5308  ccom 5314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-xp 5316  df-co 5319
This theorem is referenced by:  xptrrel  14059  relexpxpnnidm  38766
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