Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018)
Ref | Expression | ||
---|---|---|---|
Assertion | relcnveq | ⊢ ( Rel 𝑅 → ( ◡ 𝑅 ⊆ 𝑅 ↔ ◡ 𝑅 = 𝑅 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnveq3 | ⊢ ( Rel 𝑅 → ( 𝑅 = ◡ 𝑅 ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 𝑅 𝑥 ) ) ) | |
2 | cnvsym | ⊢ ( ◡ 𝑅 ⊆ 𝑅 ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 𝑅 𝑥 ) ) | |
3 | 1 2 | bitr4di | ⊢ ( Rel 𝑅 → ( 𝑅 = ◡ 𝑅 ↔ ◡ 𝑅 ⊆ 𝑅 ) ) |
4 | eqcom | ⊢ ( 𝑅 = ◡ 𝑅 ↔ ◡ 𝑅 = 𝑅 ) | |
5 | 3 4 | bitr3di | ⊢ ( Rel 𝑅 → ( ◡ 𝑅 ⊆ 𝑅 ↔ ◡ 𝑅 = 𝑅 ) ) |