Theorem xpcoidgend 14538

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 4115	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
2		xpcoidgend.1	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)
3	1, 2	eqnetrrid 3016	. 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅)
4	3	xpcogend 14537	1 ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))

Syntax hints:   → wi 4   = wceq 1543   ≠ wne 2940   ∩ cin 3865  ∅c0 4237   × cxp 5549   ∘ ccom 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-co 5560

This theorem is referenced by:  xptrrel  14543  relexpxpnnidm  40988