Step | Hyp | Ref
| Expression |
1 | | ssrel 5130 |
. . . 4
⊢ (Rel
𝐹 → (𝐹 ⊆ (V ∖ ((V ∖ I ) ∘
𝐹)) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐹 → 〈𝑥, 𝑦〉 ∈ (V ∖ ((V ∖ I )
∘ 𝐹))))) |
2 | | impexp 461 |
. . . . . . 7
⊢
(((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧) ↔ (〈𝑥, 𝑦〉 ∈ 𝐹 → (〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧))) |
3 | 2 | albii 1737 |
. . . . . 6
⊢
(∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑧(〈𝑥, 𝑦〉 ∈ 𝐹 → (〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧))) |
4 | | 19.21v 1855 |
. . . . . 6
⊢
(∀𝑧(〈𝑥, 𝑦〉 ∈ 𝐹 → (〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧)) ↔ (〈𝑥, 𝑦〉 ∈ 𝐹 → ∀𝑧(〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧))) |
5 | | vex 3176 |
. . . . . . . . . . 11
⊢ 𝑥 ∈ V |
6 | | vex 3176 |
. . . . . . . . . . 11
⊢ 𝑦 ∈ V |
7 | 5, 6 | opelco 5215 |
. . . . . . . . . 10
⊢
(〈𝑥, 𝑦〉 ∈ ((V ∖ I )
∘ 𝐹) ↔
∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦)) |
8 | | df-br 4584 |
. . . . . . . . . . . 12
⊢ (𝑥𝐹𝑧 ↔ 〈𝑥, 𝑧〉 ∈ 𝐹) |
9 | | brv 31154 |
. . . . . . . . . . . . . 14
⊢ 𝑧V𝑦 |
10 | | brdif 4635 |
. . . . . . . . . . . . . 14
⊢ (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦)) |
11 | 9, 10 | mpbiran 955 |
. . . . . . . . . . . . 13
⊢ (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦) |
12 | 6 | ideq 5196 |
. . . . . . . . . . . . . 14
⊢ (𝑧 I 𝑦 ↔ 𝑧 = 𝑦) |
13 | | equcom 1932 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧) |
14 | 12, 13 | bitri 263 |
. . . . . . . . . . . . 13
⊢ (𝑧 I 𝑦 ↔ 𝑦 = 𝑧) |
15 | 11, 14 | xchbinx 323 |
. . . . . . . . . . . 12
⊢ (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧) |
16 | 8, 15 | anbi12i 729 |
. . . . . . . . . . 11
⊢ ((𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦) ↔ (〈𝑥, 𝑧〉 ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧)) |
17 | 16 | exbii 1764 |
. . . . . . . . . 10
⊢
(∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦) ↔ ∃𝑧(〈𝑥, 𝑧〉 ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧)) |
18 | | exanali 1773 |
. . . . . . . . . 10
⊢
(∃𝑧(〈𝑥, 𝑧〉 ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧) ↔ ¬ ∀𝑧(〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧)) |
19 | 7, 17, 18 | 3bitri 285 |
. . . . . . . . 9
⊢
(〈𝑥, 𝑦〉 ∈ ((V ∖ I )
∘ 𝐹) ↔ ¬
∀𝑧(〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧)) |
20 | 19 | con2bii 346 |
. . . . . . . 8
⊢
(∀𝑧(〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧) ↔ ¬ 〈𝑥, 𝑦〉 ∈ ((V ∖ I ) ∘ 𝐹)) |
21 | | opex 4859 |
. . . . . . . . 9
⊢
〈𝑥, 𝑦〉 ∈ V |
22 | | eldif 3550 |
. . . . . . . . 9
⊢
(〈𝑥, 𝑦〉 ∈ (V ∖ ((V
∖ I ) ∘ 𝐹))
↔ (〈𝑥, 𝑦〉 ∈ V ∧ ¬
〈𝑥, 𝑦〉 ∈ ((V ∖ I ) ∘ 𝐹))) |
23 | 21, 22 | mpbiran 955 |
. . . . . . . 8
⊢
(〈𝑥, 𝑦〉 ∈ (V ∖ ((V
∖ I ) ∘ 𝐹))
↔ ¬ 〈𝑥, 𝑦〉 ∈ ((V ∖ I )
∘ 𝐹)) |
24 | 20, 23 | bitr4i 266 |
. . . . . . 7
⊢
(∀𝑧(〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧) ↔ 〈𝑥, 𝑦〉 ∈ (V ∖ ((V ∖ I )
∘ 𝐹))) |
25 | 24 | imbi2i 325 |
. . . . . 6
⊢
((〈𝑥, 𝑦〉 ∈ 𝐹 → ∀𝑧(〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧)) ↔ (〈𝑥, 𝑦〉 ∈ 𝐹 → 〈𝑥, 𝑦〉 ∈ (V ∖ ((V ∖ I )
∘ 𝐹)))) |
26 | 3, 4, 25 | 3bitri 285 |
. . . . 5
⊢
(∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧) ↔ (〈𝑥, 𝑦〉 ∈ 𝐹 → 〈𝑥, 𝑦〉 ∈ (V ∖ ((V ∖ I )
∘ 𝐹)))) |
27 | 26 | 2albii 1738 |
. . . 4
⊢
(∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐹 → 〈𝑥, 𝑦〉 ∈ (V ∖ ((V ∖ I )
∘ 𝐹)))) |
28 | 1, 27 | syl6rbbr 278 |
. . 3
⊢ (Rel
𝐹 → (∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧) ↔ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘
𝐹)))) |
29 | 28 | pm5.32i 667 |
. 2
⊢ ((Rel
𝐹 ∧ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹 ∧ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘
𝐹)))) |
30 | | dffun4 5816 |
. 2
⊢ (Fun
𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧))) |
31 | | sscoid 31190 |
. 2
⊢ (𝐹 ⊆ ( I ∘ (V ∖
((V ∖ I ) ∘ 𝐹))) ↔ (Rel 𝐹 ∧ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘
𝐹)))) |
32 | 29, 30, 31 | 3bitr4i 291 |
1
⊢ (Fun
𝐹 ↔ 𝐹 ⊆ ( I ∘ (V ∖ ((V ∖
I ) ∘ 𝐹)))) |