Metamath Proof Explorer


Theorem dfcoss3

Description: Alternate definition of the class of cosets by R (see the comment of df-coss ). (Contributed by Peter Mazsa, 27-Dec-2018)

Ref Expression
Assertion dfcoss3 𝑅 = ( 𝑅 𝑅 )

Proof

Step Hyp Ref Expression
1 brcnvg ( ( 𝑥 ∈ V ∧ 𝑢 ∈ V ) → ( 𝑥 𝑅 𝑢𝑢 𝑅 𝑥 ) )
2 1 el2v ( 𝑥 𝑅 𝑢𝑢 𝑅 𝑥 )
3 2 anbi1i ( ( 𝑥 𝑅 𝑢𝑢 𝑅 𝑦 ) ↔ ( 𝑢 𝑅 𝑥𝑢 𝑅 𝑦 ) )
4 3 exbii ( ∃ 𝑢 ( 𝑥 𝑅 𝑢𝑢 𝑅 𝑦 ) ↔ ∃ 𝑢 ( 𝑢 𝑅 𝑥𝑢 𝑅 𝑦 ) )
5 4 opabbii { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑥 𝑅 𝑢𝑢 𝑅 𝑦 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥𝑢 𝑅 𝑦 ) }
6 df-co ( 𝑅 𝑅 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑥 𝑅 𝑢𝑢 𝑅 𝑦 ) }
7 df-coss 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥𝑢 𝑅 𝑦 ) }
8 5 6 7 3eqtr4ri 𝑅 = ( 𝑅 𝑅 )