Metamath Proof Explorer


Theorem elrels5

Description: Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021)

Ref Expression
Assertion elrels5 ( 𝑅𝑉 → ( 𝑅 ∈ Rels ↔ ( 𝑅 ↾ dom 𝑅 ) = 𝑅 ) )

Proof

Step Hyp Ref Expression
1 elrelsrel ( 𝑅𝑉 → ( 𝑅 ∈ Rels ↔ Rel 𝑅 ) )
2 dfrel5 ( Rel 𝑅 ↔ ( 𝑅 ↾ dom 𝑅 ) = 𝑅 )
3 1 2 bitrdi ( 𝑅𝑉 → ( 𝑅 ∈ Rels ↔ ( 𝑅 ↾ dom 𝑅 ) = 𝑅 ) )