This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem brtpos

Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015)

Ref Expression
Assertion brtpos ( 𝐶 ∈ 𝑉 → ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝐶 ↔ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ) )

Proof

Step Hyp Ref Expression
1 brtpos2 ⊢ ( 𝐶 ∈ 𝑉 → ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝐶 ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ◡ dom 𝐹 ∪ { ∅ } ) ∧ ∪ ◡ { ⟨ 𝐴 , 𝐵 ⟩ } 𝐹 𝐶 ) ) )
2 1 adantr ⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝐶 ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ◡ dom 𝐹 ∪ { ∅ } ) ∧ ∪ ◡ { ⟨ 𝐴 , 𝐵 ⟩ } 𝐹 𝐶 ) ) )
3 opex ⊢ ⟨ 𝐵 , 𝐴 ⟩ ∈ V
4 breldmg ⊢ ( ( ⟨ 𝐵 , 𝐴 ⟩ ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ) → ⟨ 𝐵 , 𝐴 ⟩ ∈ dom 𝐹 )
5 4 3expia ⊢ ( ( ⟨ 𝐵 , 𝐴 ⟩ ∈ V ∧ 𝐶 ∈ 𝑉 ) → ( ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 → ⟨ 𝐵 , 𝐴 ⟩ ∈ dom 𝐹 ) )
6 3 5 mpan ⊢ ( 𝐶 ∈ 𝑉 → ( ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 → ⟨ 𝐵 , 𝐴 ⟩ ∈ dom 𝐹 ) )
7 6 adantr ⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ( ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 → ⟨ 𝐵 , 𝐴 ⟩ ∈ dom 𝐹 ) )
8 opelcnvg ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ◡ dom 𝐹 ↔ ⟨ 𝐵 , 𝐴 ⟩ ∈ dom 𝐹 ) )
9 8 adantl ⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ◡ dom 𝐹 ↔ ⟨ 𝐵 , 𝐴 ⟩ ∈ dom 𝐹 ) )
10 7 9 sylibrd ⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ( ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 → ⟨ 𝐴 , 𝐵 ⟩ ∈ ◡ dom 𝐹 ) )
11 elun1 ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ◡ dom 𝐹 → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ◡ dom 𝐹 ∪ { ∅ } ) )
12 10 11 syl6 ⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ( ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ◡ dom 𝐹 ∪ { ∅ } ) ) )
13 12 pm4.71rd ⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ( ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ◡ dom 𝐹 ∪ { ∅ } ) ∧ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ) ) )
14 opswap ⊢ ∪ ◡ { ⟨ 𝐴 , 𝐵 ⟩ } = ⟨ 𝐵 , 𝐴 ⟩
15 14 breq1i ⊢ ( ∪ ◡ { ⟨ 𝐴 , 𝐵 ⟩ } 𝐹 𝐶 ↔ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 )
16 15 anbi2i ⊢ ( ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ◡ dom 𝐹 ∪ { ∅ } ) ∧ ∪ ◡ { ⟨ 𝐴 , 𝐵 ⟩ } 𝐹 𝐶 ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ◡ dom 𝐹 ∪ { ∅ } ) ∧ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ) )
17 13 16 bitr4di ⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ( ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ◡ dom 𝐹 ∪ { ∅ } ) ∧ ∪ ◡ { ⟨ 𝐴 , 𝐵 ⟩ } 𝐹 𝐶 ) ) )
18 2 17 bitr4d ⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝐶 ↔ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ) )
19 18 ex ⊢ ( 𝐶 ∈ 𝑉 → ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝐶 ↔ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ) ) )
20 brtpos0 ⊢ ( 𝐶 ∈ 𝑉 → ( ∅ tpos 𝐹 𝐶 ↔ ∅ 𝐹 𝐶 ) )
21 opprc ⊢ ( ¬ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ⟨ 𝐴 , 𝐵 ⟩ = ∅ )
22 21 breq1d ⊢ ( ¬ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝐶 ↔ ∅ tpos 𝐹 𝐶 ) )
23 ancom ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ↔ ( 𝐵 ∈ V ∧ 𝐴 ∈ V ) )
24 opprc ⊢ ( ¬ ( 𝐵 ∈ V ∧ 𝐴 ∈ V ) → ⟨ 𝐵 , 𝐴 ⟩ = ∅ )
25 24 breq1d ⊢ ( ¬ ( 𝐵 ∈ V ∧ 𝐴 ∈ V ) → ( ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ↔ ∅ 𝐹 𝐶 ) )
26 23 25 sylnbi ⊢ ( ¬ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ↔ ∅ 𝐹 𝐶 ) )
27 22 26 bibi12d ⊢ ( ¬ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝐶 ↔ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ) ↔ ( ∅ tpos 𝐹 𝐶 ↔ ∅ 𝐹 𝐶 ) ) )
28 20 27 syl5ibrcom ⊢ ( 𝐶 ∈ 𝑉 → ( ¬ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝐶 ↔ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ) ) )
29 19 28 pm2.61d ⊢ ( 𝐶 ∈ 𝑉 → ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝐶 ↔ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝐶 ) )