Step | Hyp | Ref
| Expression |
1 | | vex 2740 |
. . . . 5
⊢ 𝑦 ∈ V |
2 | 1 | opelres 4912 |
. . . 4
⊢
(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑥 ∈ dom 𝐺)) |
3 | | ssel 3149 |
. . . . . . 7
⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐹)) |
4 | | vex 2740 |
. . . . . . . . 9
⊢ 𝑥 ∈ V |
5 | 4, 1 | opeldm 4830 |
. . . . . . . 8
⊢
(⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺) |
6 | 5 | a1i 9 |
. . . . . . 7
⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺)) |
7 | 3, 6 | jcad 307 |
. . . . . 6
⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑥 ∈ dom 𝐺))) |
8 | 7 | adantl 277 |
. . . . 5
⊢ ((Fun
𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑥 ∈ dom 𝐺))) |
9 | | funeu2 5242 |
. . . . . . . . . . . 12
⊢ ((Fun
𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ∃!𝑦⟨𝑥, 𝑦⟩ ∈ 𝐹) |
10 | 4 | eldm2 4825 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ dom 𝐺 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐺) |
11 | 3 | ancrd 326 |
. . . . . . . . . . . . . . 15
⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺))) |
12 | 11 | eximdv 1880 |
. . . . . . . . . . . . . 14
⊢ (𝐺 ⊆ 𝐹 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺))) |
13 | 10, 12 | biimtrid 152 |
. . . . . . . . . . . . 13
⊢ (𝐺 ⊆ 𝐹 → (𝑥 ∈ dom 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺))) |
14 | 13 | imp 124 |
. . . . . . . . . . . 12
⊢ ((𝐺 ⊆ 𝐹 ∧ 𝑥 ∈ dom 𝐺) → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)) |
15 | | eupick 2105 |
. . . . . . . . . . . 12
⊢
((∃!𝑦⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)) |
16 | 9, 14, 15 | syl2an 289 |
. . . . . . . . . . 11
⊢ (((Fun
𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺 ⊆ 𝐹 ∧ 𝑥 ∈ dom 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)) |
17 | 16 | exp43 372 |
. . . . . . . . . 10
⊢ (Fun
𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝐺 ⊆ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))) |
18 | 17 | com23 78 |
. . . . . . . . 9
⊢ (Fun
𝐹 → (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))) |
19 | 18 | imp 124 |
. . . . . . . 8
⊢ ((Fun
𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))) |
20 | 19 | com34 83 |
. . . . . . 7
⊢ ((Fun
𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))) |
21 | 20 | pm2.43d 50 |
. . . . . 6
⊢ ((Fun
𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))) |
22 | 21 | impd 254 |
. . . . 5
⊢ ((Fun
𝐹 ∧ 𝐺 ⊆ 𝐹) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑥 ∈ dom 𝐺) → ⟨𝑥, 𝑦⟩ ∈ 𝐺)) |
23 | 8, 22 | impbid 129 |
. . . 4
⊢ ((Fun
𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑥 ∈ dom 𝐺))) |
24 | 2, 23 | bitr4id 199 |
. . 3
⊢ ((Fun
𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)) |
25 | 24 | alrimivv 1875 |
. 2
⊢ ((Fun
𝐹 ∧ 𝐺 ⊆ 𝐹) → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)) |
26 | | relres 4935 |
. . 3
⊢ Rel
(𝐹 ↾ dom 𝐺) |
27 | | funrel 5233 |
. . . 4
⊢ (Fun
𝐹 → Rel 𝐹) |
28 | | relss 4713 |
. . . 4
⊢ (𝐺 ⊆ 𝐹 → (Rel 𝐹 → Rel 𝐺)) |
29 | 27, 28 | mpan9 281 |
. . 3
⊢ ((Fun
𝐹 ∧ 𝐺 ⊆ 𝐹) → Rel 𝐺) |
30 | | eqrel 4715 |
. . 3
⊢ ((Rel
(𝐹 ↾ dom 𝐺) ∧ Rel 𝐺) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))) |
31 | 26, 29, 30 | sylancr 414 |
. 2
⊢ ((Fun
𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))) |
32 | 25, 31 | mpbird 167 |
1
⊢ ((Fun
𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺) |