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Metamath Proof Explorer

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 ) ) ) )