Metamath Proof Explorer


Theorem dfcoss4

Description: Alternate definition of the class of cosets by R (see the comment of df-coss ). (Contributed by Peter Mazsa, 12-Jul-2021)

Ref Expression
Assertion dfcoss4 𝑅 = ran ( 𝑅𝑅 )

Proof

Step Hyp Ref Expression
1 df-coss 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥𝑢 𝑅 𝑦 ) }
2 rnxrn ran ( 𝑅𝑅 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥𝑢 𝑅 𝑦 ) }
3 1 2 eqtr4i 𝑅 = ran ( 𝑅𝑅 )