Step | Hyp | Ref
| Expression |
1 | | elfv 5176 |
. . 3
⊢ (𝑥 ∈ (𝐹‘𝐴) ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))) |
2 | | bi2 121 |
. . . . . . . . . 10
⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝑦 = 𝑧 → 𝐴𝐹𝑦)) |
3 | 2 | alimi 1344 |
. . . . . . . . 9
⊢
(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ∀𝑦(𝑦 = 𝑧 → 𝐴𝐹𝑦)) |
4 | | vex 2560 |
. . . . . . . . . 10
⊢ 𝑧 ∈ V |
5 | | breq2 3768 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑧 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝑧)) |
6 | 4, 5 | ceqsalv 2584 |
. . . . . . . . 9
⊢
(∀𝑦(𝑦 = 𝑧 → 𝐴𝐹𝑦) ↔ 𝐴𝐹𝑧) |
7 | 3, 6 | sylib 127 |
. . . . . . . 8
⊢
(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → 𝐴𝐹𝑧) |
8 | 7 | anim2i 324 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → (𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧)) |
9 | 8 | eximi 1491 |
. . . . . 6
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧)) |
10 | | elequ2 1601 |
. . . . . . . 8
⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦)) |
11 | | breq2 3768 |
. . . . . . . 8
⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦)) |
12 | 10, 11 | anbi12d 442 |
. . . . . . 7
⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦))) |
13 | 12 | cbvexv 1795 |
. . . . . 6
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦)) |
14 | 9, 13 | sylib 127 |
. . . . 5
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦)) |
15 | | exsimpr 1509 |
. . . . . 6
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) |
16 | | df-eu 1903 |
. . . . . 6
⊢
(∃!𝑦 𝐴𝐹𝑦 ↔ ∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) |
17 | 15, 16 | sylibr 137 |
. . . . 5
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃!𝑦 𝐴𝐹𝑦) |
18 | 14, 17 | jca 290 |
. . . 4
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)) |
19 | | nfeu1 1911 |
. . . . . . 7
⊢
Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦 |
20 | | nfv 1421 |
. . . . . . . . 9
⊢
Ⅎ𝑦 𝑥 ∈ 𝑧 |
21 | | nfa1 1434 |
. . . . . . . . 9
⊢
Ⅎ𝑦∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) |
22 | 20, 21 | nfan 1457 |
. . . . . . . 8
⊢
Ⅎ𝑦(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) |
23 | 22 | nfex 1528 |
. . . . . . 7
⊢
Ⅎ𝑦∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) |
24 | 19, 23 | nfim 1464 |
. . . . . 6
⊢
Ⅎ𝑦(∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))) |
25 | | bi1 111 |
. . . . . . . . . . . . . 14
⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝐴𝐹𝑦 → 𝑦 = 𝑧)) |
26 | | ax-14 1405 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧)) |
27 | 25, 26 | syl6 29 |
. . . . . . . . . . . . 13
⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝐴𝐹𝑦 → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧))) |
28 | 27 | com23 72 |
. . . . . . . . . . . 12
⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝑥 ∈ 𝑦 → (𝐴𝐹𝑦 → 𝑥 ∈ 𝑧))) |
29 | 28 | impd 242 |
. . . . . . . . . . 11
⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → 𝑥 ∈ 𝑧)) |
30 | 29 | sps 1430 |
. . . . . . . . . 10
⊢
(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → 𝑥 ∈ 𝑧)) |
31 | 30 | anc2ri 313 |
. . . . . . . . 9
⊢
(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))) |
32 | 31 | com12 27 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))) |
33 | 32 | eximdv 1760 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))) |
34 | 16, 33 | syl5bi 141 |
. . . . . 6
⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))) |
35 | 24, 34 | exlimi 1485 |
. . . . 5
⊢
(∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))) |
36 | 35 | imp 115 |
. . . 4
⊢
((∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦) → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))) |
37 | 18, 36 | impbii 117 |
. . 3
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)) |
38 | 1, 37 | bitri 173 |
. 2
⊢ (𝑥 ∈ (𝐹‘𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)) |
39 | 38 | abbi2i 2152 |
1
⊢ (𝐹‘𝐴) = {𝑥 ∣ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)} |