Step | Hyp | Ref
| Expression |
1 | | opswapg 5116 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩) |
2 | 1 | breq1d 4014 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)) |
3 | 2 | 3adant3 1017 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)) |
4 | 3 | anbi2d 464 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶) ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴⟩𝐹𝐶))) |
5 | | brtpos2 6252 |
. . 3
⊢ (𝐶 ∈ 𝑋 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶))) |
6 | 5 | 3ad2ant3 1020 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{⟨𝐴, 𝐵⟩}𝐹𝐶))) |
7 | | opexg 4229 |
. . . . . . . . 9
⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ⟨𝐵, 𝐴⟩ ∈ V) |
8 | 7 | ancoms 268 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐵, 𝐴⟩ ∈ V) |
9 | 8 | anim1i 340 |
. . . . . . 7
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑋)) |
10 | 9 | 3impa 1194 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑋)) |
11 | | breldmg 4834 |
. . . . . . 7
⊢
((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑋 ∧ ⟨𝐵, 𝐴⟩𝐹𝐶) → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹) |
12 | 11 | 3expia 1205 |
. . . . . 6
⊢
((⟨𝐵, 𝐴⟩ ∈ V ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)) |
13 | 10, 12 | syl 14 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)) |
14 | | opelcnvg 4808 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)) |
15 | 14 | 3adant3 1017 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)) |
16 | 13, 15 | sylibrd 169 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹)) |
17 | | elun1 3303 |
. . . 4
⊢
(⟨𝐴, 𝐵⟩ ∈ ◡dom 𝐹 → ⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅})) |
18 | 16, 17 | syl6 33 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩𝐹𝐶 → ⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}))) |
19 | 18 | pm4.71rd 394 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ⟨𝐵, 𝐴⟩𝐹𝐶))) |
20 | 4, 6, 19 | 3bitr4d 220 |
1
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)) |