Description: Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | dfsymrels3 | ⊢ SymRels = { 𝑟 ∈ Rels ∣ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑟 𝑦 → 𝑦 𝑟 𝑥 ) } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsymrels2 | ⊢ SymRels = { 𝑟 ∈ Rels ∣ ◡ 𝑟 ⊆ 𝑟 } | |
2 | cnvsym | ⊢ ( ◡ 𝑟 ⊆ 𝑟 ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑟 𝑦 → 𝑦 𝑟 𝑥 ) ) | |
3 | 1 2 | rabbieq | ⊢ SymRels = { 𝑟 ∈ Rels ∣ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑟 𝑦 → 𝑦 𝑟 𝑥 ) } |