Metamath Proof Explorer


Theorem dfsymrels3

Description: Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021)

Ref Expression
Assertion dfsymrels3 SymRels = { 𝑟 ∈ Rels ∣ ∀ 𝑥𝑦 ( 𝑥 𝑟 𝑦𝑦 𝑟 𝑥 ) }

Proof

Step Hyp Ref Expression
1 dfsymrels2 SymRels = { 𝑟 ∈ Rels ∣ 𝑟𝑟 }
2 cnvsym ( 𝑟𝑟 ↔ ∀ 𝑥𝑦 ( 𝑥 𝑟 𝑦𝑦 𝑟 𝑥 ) )
3 1 2 rabbieq SymRels = { 𝑟 ∈ Rels ∣ ∀ 𝑥𝑦 ( 𝑥 𝑟 𝑦𝑦 𝑟 𝑥 ) }