Step | Hyp | Ref
| Expression |
1 | | impexp 452 |
. . . . . . 7
⊢
(((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧))) |
2 | 1 | albii 1822 |
. . . . . 6
⊢
(∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧))) |
3 | | 19.21v 1943 |
. . . . . 6
⊢
(∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧))) |
4 | | vex 3451 |
. . . . . . . . . . 11
⊢ 𝑥 ∈ V |
5 | | vex 3451 |
. . . . . . . . . . 11
⊢ 𝑦 ∈ V |
6 | 4, 5 | opelco 5831 |
. . . . . . . . . 10
⊢
(⟨𝑥, 𝑦⟩ ∈ ((V ∖ I )
∘ 𝐹) ↔
∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦)) |
7 | | df-br 5110 |
. . . . . . . . . . . 12
⊢ (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹) |
8 | | brv 5433 |
. . . . . . . . . . . . . 14
⊢ 𝑧V𝑦 |
9 | | brdif 5162 |
. . . . . . . . . . . . . 14
⊢ (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦)) |
10 | 8, 9 | mpbiran 708 |
. . . . . . . . . . . . 13
⊢ (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦) |
11 | 5 | ideq 5812 |
. . . . . . . . . . . . . 14
⊢ (𝑧 I 𝑦 ↔ 𝑧 = 𝑦) |
12 | | equcom 2022 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧) |
13 | 11, 12 | bitri 275 |
. . . . . . . . . . . . 13
⊢ (𝑧 I 𝑦 ↔ 𝑦 = 𝑧) |
14 | 10, 13 | xchbinx 334 |
. . . . . . . . . . . 12
⊢ (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧) |
15 | 7, 14 | anbi12i 628 |
. . . . . . . . . . 11
⊢ ((𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦) ↔ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧)) |
16 | 15 | exbii 1851 |
. . . . . . . . . 10
⊢
(∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦) ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧)) |
17 | | exanali 1863 |
. . . . . . . . . 10
⊢
(∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧) ↔ ¬ ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)) |
18 | 6, 16, 17 | 3bitri 297 |
. . . . . . . . 9
⊢
(⟨𝑥, 𝑦⟩ ∈ ((V ∖ I )
∘ 𝐹) ↔ ¬
∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)) |
19 | 18 | con2bii 358 |
. . . . . . . 8
⊢
(∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹)) |
20 | | opex 5425 |
. . . . . . . . 9
⊢
⟨𝑥, 𝑦⟩ ∈ V |
21 | | eldif 3924 |
. . . . . . . . 9
⊢
(⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V
∖ I ) ∘ 𝐹))
↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬
⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹))) |
22 | 20, 21 | mpbiran 708 |
. . . . . . . 8
⊢
(⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V
∖ I ) ∘ 𝐹))
↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I )
∘ 𝐹)) |
23 | 19, 22 | bitr4i 278 |
. . . . . . 7
⊢
(∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧) ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I )
∘ 𝐹))) |
24 | 23 | imbi2i 336 |
. . . . . 6
⊢
((⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I )
∘ 𝐹)))) |
25 | 2, 3, 24 | 3bitri 297 |
. . . . 5
⊢
(∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I )
∘ 𝐹)))) |
26 | 25 | 2albii 1823 |
. . . 4
⊢
(∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I )
∘ 𝐹)))) |
27 | | ssrel 5742 |
. . . 4
⊢ (Rel
𝐹 → (𝐹 ⊆ (V ∖ ((V ∖ I ) ∘
𝐹)) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I )
∘ 𝐹))))) |
28 | 26, 27 | bitr4id 290 |
. . 3
⊢ (Rel
𝐹 → (∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘
𝐹)))) |
29 | 28 | pm5.32i 576 |
. 2
⊢ ((Rel
𝐹 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹 ∧ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘
𝐹)))) |
30 | | dffun4 6516 |
. 2
⊢ (Fun
𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))) |
31 | | sscoid 34551 |
. 2
⊢ (𝐹 ⊆ ( I ∘ (V ∖
((V ∖ I ) ∘ 𝐹))) ↔ (Rel 𝐹 ∧ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘
𝐹)))) |
32 | 29, 30, 31 | 3bitr4i 303 |
1
⊢ (Fun
𝐹 ↔ 𝐹 ⊆ ( I ∘ (V ∖ ((V ∖
I ) ∘ 𝐹)))) |