Theorem br1cossincnvepres

Description: B and C are cosets by an intersection with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021)

	Ref	Expression
Assertion	br1cossincnvepres	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ≀ ( 𝑅 ∩ ( ◡ E ↾ 𝐴 ) ) 𝐶 ↔ ∃ 𝑢 ∈ 𝐴 ( ( 𝐵 ∈ 𝑢 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝐶 ∈ 𝑢 ∧ 𝑢 𝑅 𝐶 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		br1cossinres	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ≀ ( 𝑅 ∩ ( ◡ E ↾ 𝐴 ) ) 𝐶 ↔ ∃ 𝑢 ∈ 𝐴 ( ( 𝑢 ◡ E 𝐵 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝑢 ◡ E 𝐶 ∧ 𝑢 𝑅 𝐶 ) ) ) )
2		brcnvep	⊢ ( 𝑢 ∈ V → ( 𝑢 ◡ E 𝐵 ↔ 𝐵 ∈ 𝑢 ) )
3	2	elv	⊢ ( 𝑢 ◡ E 𝐵 ↔ 𝐵 ∈ 𝑢 )
4	3	anbi1i	⊢ ( ( 𝑢 ◡ E 𝐵 ∧ 𝑢 𝑅 𝐵 ) ↔ ( 𝐵 ∈ 𝑢 ∧ 𝑢 𝑅 𝐵 ) )
5		brcnvep	⊢ ( 𝑢 ∈ V → ( 𝑢 ◡ E 𝐶 ↔ 𝐶 ∈ 𝑢 ) )
6	5	elv	⊢ ( 𝑢 ◡ E 𝐶 ↔ 𝐶 ∈ 𝑢 )
7	6	anbi1i	⊢ ( ( 𝑢 ◡ E 𝐶 ∧ 𝑢 𝑅 𝐶 ) ↔ ( 𝐶 ∈ 𝑢 ∧ 𝑢 𝑅 𝐶 ) )
8	4 7	anbi12i	⊢ ( ( ( 𝑢 ◡ E 𝐵 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝑢 ◡ E 𝐶 ∧ 𝑢 𝑅 𝐶 ) ) ↔ ( ( 𝐵 ∈ 𝑢 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝐶 ∈ 𝑢 ∧ 𝑢 𝑅 𝐶 ) ) )
9	8	rexbii	⊢ ( ∃ 𝑢 ∈ 𝐴 ( ( 𝑢 ◡ E 𝐵 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝑢 ◡ E 𝐶 ∧ 𝑢 𝑅 𝐶 ) ) ↔ ∃ 𝑢 ∈ 𝐴 ( ( 𝐵 ∈ 𝑢 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝐶 ∈ 𝑢 ∧ 𝑢 𝑅 𝐶 ) ) )
10	1 9	bitrdi	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ≀ ( 𝑅 ∩ ( ◡ E ↾ 𝐴 ) ) 𝐶 ↔ ∃ 𝑢 ∈ 𝐴 ( ( 𝐵 ∈ 𝑢 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝐶 ∈ 𝑢 ∧ 𝑢 𝑅 𝐶 ) ) ) )