Metamath Proof Explorer


Theorem relcnveq

Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018)

Ref Expression
Assertion relcnveq ( Rel 𝑅 → ( 𝑅𝑅 𝑅 = 𝑅 ) )

Proof

Step Hyp Ref Expression
1 relcnveq3 ( Rel 𝑅 → ( 𝑅 = 𝑅 ↔ ∀ 𝑥𝑦 ( 𝑥 𝑅 𝑦𝑦 𝑅 𝑥 ) ) )
2 cnvsym ( 𝑅𝑅 ↔ ∀ 𝑥𝑦 ( 𝑥 𝑅 𝑦𝑦 𝑅 𝑥 ) )
3 1 2 bitr4di ( Rel 𝑅 → ( 𝑅 = 𝑅 𝑅𝑅 ) )
4 eqcom ( 𝑅 = 𝑅 𝑅 = 𝑅 )
5 3 4 bitr3di ( Rel 𝑅 → ( 𝑅𝑅 𝑅 = 𝑅 ) )