Theorem br1cossres

Description: B and C are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 30-Dec-2018)

	Ref	Expression
Assertion	br1cossres	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ≀ ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝐵 ∧ 𝑢 𝑅 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		brcoss	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ≀ ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ∃ 𝑢 ( 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ∧ 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐶 ) ) )
2		exanres	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ∃ 𝑢 ( 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ∧ 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐶 ) ↔ ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝐵 ∧ 𝑢 𝑅 𝐶 ) ) )
3	1 2	bitrd	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ≀ ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝐵 ∧ 𝑢 𝑅 𝐶 ) ) )