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Metamath Proof Explorer

Theorem brcosscnv

Description: A and B are cosets by converse R : a binary relation. (Contributed by Peter Mazsa, 23-Jan-2019)

Ref Expression
Assertion brcosscnv ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≀ ◡ 𝑅 𝐵 ↔ ∃ 𝑥 ( 𝐴 𝑅 𝑥 ∧ 𝐵 𝑅 𝑥 ) ) )

Proof

Step Hyp Ref Expression
1 brcoss ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≀ ◡ 𝑅 𝐵 ↔ ∃ 𝑥 ( 𝑥 ◡ 𝑅 𝐴 ∧ 𝑥 ◡ 𝑅 𝐵 ) ) )
2 brcnvg ⊢ ( ( 𝑥 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝑥 ◡ 𝑅 𝐴 ↔ 𝐴 𝑅 𝑥 ) )
3 2 el2v1 ⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 ◡ 𝑅 𝐴 ↔ 𝐴 𝑅 𝑥 ) )
4 brcnvg ⊢ ( ( 𝑥 ∈ V ∧ 𝐵 ∈ 𝑊 ) → ( 𝑥 ◡ 𝑅 𝐵 ↔ 𝐵 𝑅 𝑥 ) )
5 4 el2v1 ⊢ ( 𝐵 ∈ 𝑊 → ( 𝑥 ◡ 𝑅 𝐵 ↔ 𝐵 𝑅 𝑥 ) )
6 3 5 bi2anan9 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝑥 ◡ 𝑅 𝐴 ∧ 𝑥 ◡ 𝑅 𝐵 ) ↔ ( 𝐴 𝑅 𝑥 ∧ 𝐵 𝑅 𝑥 ) ) )
7 6 exbidv ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 ( 𝑥 ◡ 𝑅 𝐴 ∧ 𝑥 ◡ 𝑅 𝐵 ) ↔ ∃ 𝑥 ( 𝐴 𝑅 𝑥 ∧ 𝐵 𝑅 𝑥 ) ) )
8 1 7 bitrd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≀ ◡ 𝑅 𝐵 ↔ ∃ 𝑥 ( 𝐴 𝑅 𝑥 ∧ 𝐵 𝑅 𝑥 ) ) )