Theorem xpcoidgend 15038

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 4165	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
2		xpcoidgend.1	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)
3	1, 2	eqnetrrid 3036	. 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅)
4	3	xpcogend 15037	1 ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2961   ∩ cin 3907  ∅c0 4289   × cxp 5664   ∘ ccom 5670

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

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 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-co 5675

This theorem is used by:  xptrrel  15043  relexpxpnnidm  44462