Metamath Proof Explorer


Theorem elsymrelsrel

Description: For sets, being an element of the class of symmetric relations ( df-symrels ) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021)

Ref Expression
Assertion elsymrelsrel ( 𝑅𝑉 → ( 𝑅 ∈ SymRels ↔ SymRel 𝑅 ) )

Proof

Step Hyp Ref Expression
1 elrelsrel ( 𝑅𝑉 → ( 𝑅 ∈ Rels ↔ Rel 𝑅 ) )
2 1 anbi2d ( 𝑅𝑉 → ( ( 𝑅𝑅𝑅 ∈ Rels ) ↔ ( 𝑅𝑅 ∧ Rel 𝑅 ) ) )
3 elsymrels2 ( 𝑅 ∈ SymRels ↔ ( 𝑅𝑅𝑅 ∈ Rels ) )
4 dfsymrel2 ( SymRel 𝑅 ↔ ( 𝑅𝑅 ∧ Rel 𝑅 ) )
5 2 3 4 3bitr4g ( 𝑅𝑉 → ( 𝑅 ∈ SymRels ↔ SymRel 𝑅 ) )