Step | Hyp | Ref
| Expression |
1 | | elfv 6737 |
. . 3
⊢ (𝑥 ∈ (𝐹‘𝐴) ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))) |
2 | | biimpr 223 |
. . . . . . . . . 10
⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝑦 = 𝑧 → 𝐴𝐹𝑦)) |
3 | 2 | alimi 1819 |
. . . . . . . . 9
⊢
(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ∀𝑦(𝑦 = 𝑧 → 𝐴𝐹𝑦)) |
4 | | breq2 5074 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑧 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝑧)) |
5 | 4 | equsalvw 2012 |
. . . . . . . . 9
⊢
(∀𝑦(𝑦 = 𝑧 → 𝐴𝐹𝑦) ↔ 𝐴𝐹𝑧) |
6 | 3, 5 | sylib 221 |
. . . . . . . 8
⊢
(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → 𝐴𝐹𝑧) |
7 | 6 | anim2i 620 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → (𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧)) |
8 | 7 | eximi 1842 |
. . . . . 6
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧)) |
9 | | elequ2 2127 |
. . . . . . . 8
⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦)) |
10 | | breq2 5074 |
. . . . . . . 8
⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦)) |
11 | 9, 10 | anbi12d 634 |
. . . . . . 7
⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦))) |
12 | 11 | cbvexvw 2045 |
. . . . . 6
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦)) |
13 | 8, 12 | sylib 221 |
. . . . 5
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦)) |
14 | | exsimpr 1877 |
. . . . . 6
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) |
15 | | eu6 2575 |
. . . . . 6
⊢
(∃!𝑦 𝐴𝐹𝑦 ↔ ∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) |
16 | 14, 15 | sylibr 237 |
. . . . 5
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃!𝑦 𝐴𝐹𝑦) |
17 | 13, 16 | jca 515 |
. . . 4
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)) |
18 | | nfeu1 2589 |
. . . . . . 7
⊢
Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦 |
19 | | nfv 1922 |
. . . . . . . . 9
⊢
Ⅎ𝑦 𝑥 ∈ 𝑧 |
20 | | nfa1 2154 |
. . . . . . . . 9
⊢
Ⅎ𝑦∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) |
21 | 19, 20 | nfan 1907 |
. . . . . . . 8
⊢
Ⅎ𝑦(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) |
22 | 21 | nfex 2325 |
. . . . . . 7
⊢
Ⅎ𝑦∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) |
23 | 18, 22 | nfim 1904 |
. . . . . 6
⊢
Ⅎ𝑦(∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))) |
24 | | biimp 218 |
. . . . . . . . . . . . 13
⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝐴𝐹𝑦 → 𝑦 = 𝑧)) |
25 | | ax9 2126 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧)) |
26 | 24, 25 | syl6 35 |
. . . . . . . . . . . 12
⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝐴𝐹𝑦 → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧))) |
27 | 26 | impcomd 415 |
. . . . . . . . . . 11
⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → 𝑥 ∈ 𝑧)) |
28 | 27 | sps 2184 |
. . . . . . . . . 10
⊢
(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → 𝑥 ∈ 𝑧)) |
29 | 28 | anc2ri 560 |
. . . . . . . . 9
⊢
(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))) |
30 | 29 | com12 32 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))) |
31 | 30 | eximdv 1925 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))) |
32 | 15, 31 | syl5bi 245 |
. . . . . 6
⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))) |
33 | 23, 32 | exlimi 2217 |
. . . . 5
⊢
(∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))) |
34 | 33 | imp 410 |
. . . 4
⊢
((∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦) → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))) |
35 | 17, 34 | impbii 212 |
. . 3
⊢
(∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)) |
36 | 1, 35 | bitri 278 |
. 2
⊢ (𝑥 ∈ (𝐹‘𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)) |
37 | 36 | abbi2i 2879 |
1
⊢ (𝐹‘𝐴) = {𝑥 ∣ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)} |