Metamath Proof Explorer


Theorem relbrcnvg

Description: When R is a relation, the sethood assumptions on brcnv can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion relbrcnvg ( Rel 𝑅 → ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐴 ) )

Proof

Step Hyp Ref Expression
1 relcnv Rel 𝑅
2 1 brrelex12i ( 𝐴 𝑅 𝐵 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
3 2 a1i ( Rel 𝑅 → ( 𝐴 𝑅 𝐵 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) )
4 brrelex12 ( ( Rel 𝑅𝐵 𝑅 𝐴 ) → ( 𝐵 ∈ V ∧ 𝐴 ∈ V ) )
5 4 ancomd ( ( Rel 𝑅𝐵 𝑅 𝐴 ) → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
6 5 ex ( Rel 𝑅 → ( 𝐵 𝑅 𝐴 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) )
7 brcnvg ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐴 ) )
8 7 a1i ( Rel 𝑅 → ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐴 ) ) )
9 3 6 8 pm5.21ndd ( Rel 𝑅 → ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐴 ) )