Theorem relbrcoss

Description: A and B are cosets by relation R : a binary relation. (Contributed by Peter Mazsa, 22-Apr-2021)

	Ref	Expression
Assertion	relbrcoss	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( Rel 𝑅 → ( 𝐴 ≀ 𝑅 𝐵 ↔ ∃ 𝑥 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑥 ] 𝑅 ∧ 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		resdm	⊢ ( Rel 𝑅 → ( 𝑅 ↾ dom 𝑅 ) = 𝑅 )
2	1	cosseqd	⊢ ( Rel 𝑅 → ≀ ( 𝑅 ↾ dom 𝑅 ) = ≀ 𝑅 )
3	2	breqd	⊢ ( Rel 𝑅 → ( 𝐴 ≀ ( 𝑅 ↾ dom 𝑅 ) 𝐵 ↔ 𝐴 ≀ 𝑅 𝐵 ) )
4	3	adantl	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Rel 𝑅 ) → ( 𝐴 ≀ ( 𝑅 ↾ dom 𝑅 ) 𝐵 ↔ 𝐴 ≀ 𝑅 𝐵 ) )
5		br1cossres2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≀ ( 𝑅 ↾ dom 𝑅 ) 𝐵 ↔ ∃ 𝑥 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑥 ] 𝑅 ∧ 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) )
6	5	adantr	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Rel 𝑅 ) → ( 𝐴 ≀ ( 𝑅 ↾ dom 𝑅 ) 𝐵 ↔ ∃ 𝑥 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑥 ] 𝑅 ∧ 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) )
7	4 6	bitr3d	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Rel 𝑅 ) → ( 𝐴 ≀ 𝑅 𝐵 ↔ ∃ 𝑥 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑥 ] 𝑅 ∧ 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) )
8	7	ex	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( Rel 𝑅 → ( 𝐴 ≀ 𝑅 𝐵 ↔ ∃ 𝑥 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑥 ] 𝑅 ∧ 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) ) )