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 𝐵 ↔ 𝐵 𝑅 𝐴 ) ) |
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 𝑅 ∧ 𝐵 𝑅 𝐴 ) → ( 𝐵 ∈ V ∧ 𝐴 ∈ V ) ) | |
5 | 4 | ancomd | ⊢ ( ( Rel 𝑅 ∧ 𝐵 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 𝐵 ↔ 𝐵 𝑅 𝐴 ) ) | |
8 | 7 | a1i | ⊢ ( Rel 𝑅 → ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ◡ 𝑅 𝐵 ↔ 𝐵 𝑅 𝐴 ) ) ) |
9 | 3 6 8 | pm5.21ndd | ⊢ ( Rel 𝑅 → ( 𝐴 ◡ 𝑅 𝐵 ↔ 𝐵 𝑅 𝐴 ) ) |