Description: Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019) (Revised by Peter Mazsa, 30-Sep-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | eqvrelcoss4 | ⊢ ( EqvRel ≀ 𝑅 ↔ ∀ 𝑥 ∀ 𝑧 ( ( [ 𝑥 ] ≀ 𝑅 ∩ [ 𝑧 ] ≀ 𝑅 ) ≠ ∅ → ( [ 𝑥 ] ◡ 𝑅 ∩ [ 𝑧 ] ◡ 𝑅 ) ≠ ∅ ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelcoss3 | ⊢ ( EqvRel ≀ 𝑅 ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 ≀ 𝑅 𝑦 ∧ 𝑦 ≀ 𝑅 𝑧 ) → 𝑥 ≀ 𝑅 𝑧 ) ) | |
2 | trcoss2 | ⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 ≀ 𝑅 𝑦 ∧ 𝑦 ≀ 𝑅 𝑧 ) → 𝑥 ≀ 𝑅 𝑧 ) ↔ ∀ 𝑥 ∀ 𝑧 ( ( [ 𝑥 ] ≀ 𝑅 ∩ [ 𝑧 ] ≀ 𝑅 ) ≠ ∅ → ( [ 𝑥 ] ◡ 𝑅 ∩ [ 𝑧 ] ◡ 𝑅 ) ≠ ∅ ) ) | |
3 | 1 2 | bitri | ⊢ ( EqvRel ≀ 𝑅 ↔ ∀ 𝑥 ∀ 𝑧 ( ( [ 𝑥 ] ≀ 𝑅 ∩ [ 𝑧 ] ≀ 𝑅 ) ≠ ∅ → ( [ 𝑥 ] ◡ 𝑅 ∩ [ 𝑧 ] ◡ 𝑅 ) ≠ ∅ ) ) |