| Step | Hyp | Ref
| Expression |
| 1 | | impexp 450 |
. . . . . . 7
⊢
(((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧) ↔ (〈𝑥, 𝑦〉 ∈ 𝐹 → (〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧))) |
| 2 | 1 | albii 1819 |
. . . . . 6
⊢
(∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑧(〈𝑥, 𝑦〉 ∈ 𝐹 → (〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧))) |
| 3 | | 19.21v 1939 |
. . . . . 6
⊢
(∀𝑧(〈𝑥, 𝑦〉 ∈ 𝐹 → (〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧)) ↔ (〈𝑥, 𝑦〉 ∈ 𝐹 → ∀𝑧(〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧))) |
| 4 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑥 ∈ V |
| 5 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑦 ∈ V |
| 6 | 4, 5 | opelco 5882 |
. . . . . . . . . 10
⊢
(〈𝑥, 𝑦〉 ∈ ((V ∖ I )
∘ 𝐹) ↔
∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦)) |
| 7 | | df-br 5144 |
. . . . . . . . . . . 12
⊢ (𝑥𝐹𝑧 ↔ 〈𝑥, 𝑧〉 ∈ 𝐹) |
| 8 | | brv 5477 |
. . . . . . . . . . . . . 14
⊢ 𝑧V𝑦 |
| 9 | | brdif 5196 |
. . . . . . . . . . . . . 14
⊢ (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦)) |
| 10 | 8, 9 | mpbiran 709 |
. . . . . . . . . . . . 13
⊢ (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦) |
| 11 | 5 | ideq 5863 |
. . . . . . . . . . . . . 14
⊢ (𝑧 I 𝑦 ↔ 𝑧 = 𝑦) |
| 12 | | equcom 2017 |
. . . . . . . . . . . . . 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 1848 |
. . . . . . . . . 10
⊢
(∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦) ↔ ∃𝑧(〈𝑥, 𝑧〉 ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧)) |
| 17 | | exanali 1859 |
. . . . . . . . . 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 357 |
. . . . . . . 8
⊢
(∀𝑧(〈𝑥, 𝑧〉 ∈ 𝐹 → 𝑦 = 𝑧) ↔ ¬ 〈𝑥, 𝑦〉 ∈ ((V ∖ I ) ∘ 𝐹)) |
| 20 | | opex 5469 |
. . . . . . . . 9
⊢
〈𝑥, 𝑦〉 ∈ V |
| 21 | | eldif 3961 |
. . . . . . . . 9
⊢
(〈𝑥, 𝑦〉 ∈ (V ∖ ((V
∖ I ) ∘ 𝐹))
↔ (〈𝑥, 𝑦〉 ∈ V ∧ ¬
〈𝑥, 𝑦〉 ∈ ((V ∖ I ) ∘ 𝐹))) |
| 22 | 20, 21 | mpbiran 709 |
. . . . . . . 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 1820 |
. . . 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 5792 |
. . . 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 574 |
. 2
⊢ ((Rel
𝐹 ∧ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹 ∧ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘
𝐹)))) |
| 30 | | dffun4 6577 |
. 2
⊢ (Fun
𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧))) |
| 31 | | sscoid 35914 |
. 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)))) |