Theorem xpcoidgend 15157

Description: If two classes are not disjoint, then the composition of their Cartesian product with itself is idempotent. (Contributed by RP, 24-Dec-2019.)

Hypothesis

Ref	Expression
xpcoidgend.1	⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)

Assertion

Ref	Expression
xpcoidgend	⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))

Proof of Theorem xpcoidgend

Step	Hyp	Ref	Expression
1		incom 4109	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
2		xpcoidgend.1	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)
3	1, 2	eqnetrrid 3034	. 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅)
4	3	xpcogend 15156	1 ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2959   ∩ cin 4092  ∅c0 4281   × cxp 5653   ∘ ccom 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-co 5664

This theorem is used by:  xptrrel  15162  relexpxpnnidm  44895