Theorem br1cossres2

Description: B and C are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 3-Jan-2018)

	Ref	Expression
Assertion	br1cossres2	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ≀ ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ∃ 𝑥 ∈ 𝐴 ( 𝐵 ∈ [ 𝑥 ] 𝑅 ∧ 𝐶 ∈ [ 𝑥 ] 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		br1cossres	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ≀ ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ∃ 𝑥 ∈ 𝐴 ( 𝑥 𝑅 𝐵 ∧ 𝑥 𝑅 𝐶 ) ) )
2		exanres3	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ∃ 𝑥 ∈ 𝐴 ( 𝐵 ∈ [ 𝑥 ] 𝑅 ∧ 𝐶 ∈ [ 𝑥 ] 𝑅 ) ↔ ∃ 𝑥 ∈ 𝐴 ( 𝑥 𝑅 𝐵 ∧ 𝑥 𝑅 𝐶 ) ) )
3	1 2	bitr4d	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ≀ ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ∃ 𝑥 ∈ 𝐴 ( 𝐵 ∈ [ 𝑥 ] 𝑅 ∧ 𝐶 ∈ [ 𝑥 ] 𝑅 ) ) )