Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | relbrtpos | ⊢ ( Rel 𝐹 → ( 〈 𝐴 , 𝐵 〉 tpos 𝐹 𝐶 ↔ 〈 𝐵 , 𝐴 〉 𝐹 𝐶 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reltpos | ⊢ Rel tpos 𝐹 | |
2 | 1 | a1i | ⊢ ( Rel 𝐹 → Rel tpos 𝐹 ) |
3 | brrelex2 | ⊢ ( ( Rel tpos 𝐹 ∧ 〈 𝐴 , 𝐵 〉 tpos 𝐹 𝐶 ) → 𝐶 ∈ V ) | |
4 | 2 3 | sylan | ⊢ ( ( Rel 𝐹 ∧ 〈 𝐴 , 𝐵 〉 tpos 𝐹 𝐶 ) → 𝐶 ∈ V ) |
5 | brrelex2 | ⊢ ( ( Rel 𝐹 ∧ 〈 𝐵 , 𝐴 〉 𝐹 𝐶 ) → 𝐶 ∈ V ) | |
6 | brtpos | ⊢ ( 𝐶 ∈ V → ( 〈 𝐴 , 𝐵 〉 tpos 𝐹 𝐶 ↔ 〈 𝐵 , 𝐴 〉 𝐹 𝐶 ) ) | |
7 | 4 5 6 | pm5.21nd | ⊢ ( Rel 𝐹 → ( 〈 𝐴 , 𝐵 〉 tpos 𝐹 𝐶 ↔ 〈 𝐵 , 𝐴 〉 𝐹 𝐶 ) ) |