Theorem xpcoidgend 14900

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 4162	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
2		xpcoidgend.1	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)
3	1, 2	eqnetrrid 3000	. 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅)
4	3	xpcogend 14899	1 ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ≠ wne 2925   ∩ cin 3904  ∅c0 4286   × cxp 5621   ∘ ccom 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-co 5632

This theorem is referenced by:  xptrrel  14905  relexpxpnnidm  43676