Step | Hyp | Ref
| Expression |
1 | | dfac3 9343 |
. 2
⊢
(CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) |
2 | | nfra1 3169 |
. . . . . 6
⊢
Ⅎ𝑧∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) |
3 | | rsp 3155 |
. . . . . . . . . . . 12
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))) |
4 | | equid 1969 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑧 = 𝑧 |
5 | | neeq1 3029 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑢 = 𝑧 → (𝑢 ≠ ∅ ↔ 𝑧 ≠ ∅)) |
6 | | eqeq1 2782 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑢 = 𝑧 → (𝑢 = 𝑧 ↔ 𝑧 = 𝑧)) |
7 | 5, 6 | anbi12d 621 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑢 = 𝑧 → ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) ↔ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧))) |
8 | 7 | rspcev 3535 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑧 ∈ 𝑥 ∧ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧)) |
9 | 4, 8 | mpanr2 691 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧)) |
10 | | fveq2 6501 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧)) |
11 | 10 | preq1d 4550 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑢 = 𝑧 → {(𝑓‘𝑢), 𝑢} = {(𝑓‘𝑧), 𝑢}) |
12 | | preq2 4545 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑢 = 𝑧 → {(𝑓‘𝑧), 𝑢} = {(𝑓‘𝑧), 𝑧}) |
13 | 11, 12 | eqtr2d 2815 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑢 = 𝑧 → {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}) |
14 | 13 | anim2i 607 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) → (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})) |
15 | 14 | reximi 3190 |
. . . . . . . . . . . . . . . . 17
⊢
(∃𝑢 ∈
𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})) |
16 | 9, 15 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})) |
17 | | prex 5190 |
. . . . . . . . . . . . . . . . 17
⊢ {(𝑓‘𝑧), 𝑧} ∈ V |
18 | | eqeq1 2782 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑔 = {(𝑓‘𝑧), 𝑧} → (𝑔 = {(𝑓‘𝑢), 𝑢} ↔ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})) |
19 | 18 | anbi2d 619 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑔 = {(𝑓‘𝑧), 𝑧} → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))) |
20 | 19 | rexbidv 3242 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑔 = {(𝑓‘𝑧), 𝑧} → (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))) |
21 | 17, 20 | elab 3582 |
. . . . . . . . . . . . . . . 16
⊢ ({(𝑓‘𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})) |
22 | 16, 21 | sylibr 226 |
. . . . . . . . . . . . . . 15
⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → {(𝑓‘𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})}) |
23 | | vex 3418 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑧 ∈ V |
24 | 23 | prid2 4574 |
. . . . . . . . . . . . . . . 16
⊢ 𝑧 ∈ {(𝑓‘𝑧), 𝑧} |
25 | | fvex 6514 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓‘𝑧) ∈ V |
26 | 25 | prid1 4573 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧} |
27 | 24, 26 | pm3.2i 463 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ {(𝑓‘𝑧), 𝑧} ∧ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧}) |
28 | | eleq2 2854 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ {(𝑓‘𝑧), 𝑧})) |
29 | | eleq2 2854 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → ((𝑓‘𝑧) ∈ 𝑣 ↔ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧})) |
30 | 28, 29 | anbi12d 621 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → ((𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣) ↔ (𝑧 ∈ {(𝑓‘𝑧), 𝑧} ∧ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧}))) |
31 | 30 | rspcev 3535 |
. . . . . . . . . . . . . . 15
⊢ (({(𝑓‘𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∧ (𝑧 ∈ {(𝑓‘𝑧), 𝑧} ∧ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧})) → ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣)) |
32 | 22, 27, 31 | sylancl 577 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣)) |
33 | | eleq1 2853 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = (𝑓‘𝑧) → (𝑤 ∈ 𝑧 ↔ (𝑓‘𝑧) ∈ 𝑧)) |
34 | | eleq1 2853 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 = (𝑓‘𝑧) → (𝑤 ∈ 𝑣 ↔ (𝑓‘𝑧) ∈ 𝑣)) |
35 | 34 | anbi2d 619 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 = (𝑓‘𝑧) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣))) |
36 | 35 | rexbidv 3242 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = (𝑓‘𝑧) → (∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣))) |
37 | 33, 36 | anbi12d 621 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = (𝑓‘𝑧) → ((𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ↔ ((𝑓‘𝑧) ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣)))) |
38 | 25, 37 | spcev 3525 |
. . . . . . . . . . . . . 14
⊢ (((𝑓‘𝑧) ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣)) → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
39 | 32, 38 | sylan2 583 |
. . . . . . . . . . . . 13
⊢ (((𝑓‘𝑧) ∈ 𝑧 ∧ (𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅)) → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
40 | 39 | ex 405 |
. . . . . . . . . . . 12
⊢ ((𝑓‘𝑧) ∈ 𝑧 → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))) |
41 | 3, 40 | syl8 76 |
. . . . . . . . . . 11
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))))) |
42 | 41 | impd 402 |
. . . . . . . . . 10
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))) |
43 | 42 | pm2.43d 53 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))) |
44 | | df-rex 3094 |
. . . . . . . . . . . . 13
⊢
(∃𝑣 ∈
{𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣(𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
45 | | vex 3418 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑣 ∈ V |
46 | | eqeq1 2782 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑔 = 𝑣 → (𝑔 = {(𝑓‘𝑢), 𝑢} ↔ 𝑣 = {(𝑓‘𝑢), 𝑢})) |
47 | 46 | anbi2d 619 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = 𝑣 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢}))) |
48 | 47 | rexbidv 3242 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑔 = 𝑣 → (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢}))) |
49 | 45, 48 | elab 3582 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢})) |
50 | | neeq1 3029 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧 = 𝑢 → (𝑧 ≠ ∅ ↔ 𝑢 ≠ ∅)) |
51 | | fveq2 6501 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑧 = 𝑢 → (𝑓‘𝑧) = (𝑓‘𝑢)) |
52 | 51 | eleq1d 2850 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑧 = 𝑢 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑧)) |
53 | | eleq2 2854 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑧 = 𝑢 → ((𝑓‘𝑢) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑢)) |
54 | 52, 53 | bitrd 271 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧 = 𝑢 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑢)) |
55 | 50, 54 | imbi12d 337 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧 = 𝑢 → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑢 ≠ ∅ → (𝑓‘𝑢) ∈ 𝑢))) |
56 | 55 | rspccv 3532 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑢 ∈ 𝑥 → (𝑢 ≠ ∅ → (𝑓‘𝑢) ∈ 𝑢))) |
57 | | elneq 8859 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑤 ∈ 𝑧 → 𝑤 ≠ 𝑧) |
58 | 57 | neneqd 2972 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑤 ∈ 𝑧 → ¬ 𝑤 = 𝑧) |
59 | | vex 3418 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ 𝑤 ∈ V |
60 | | neqne 2975 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (¬
𝑤 = 𝑧 → 𝑤 ≠ 𝑧) |
61 | | prel12g 4669 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑤 ∈ V ∧ 𝑧 ∈ V ∧ 𝑤 ≠ 𝑧) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢}))) |
62 | 59, 23, 60, 61 | mp3an12i 1444 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (¬
𝑤 = 𝑧 → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢}))) |
63 | | ancom 453 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) |
64 | | eleq2 2854 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ {(𝑓‘𝑢), 𝑢})) |
65 | | eleq2 2854 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ {(𝑓‘𝑢), 𝑢})) |
66 | 64, 65 | anbi12d 621 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣) ↔ (𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢}))) |
67 | 63, 66 | syl5rbbr 278 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢}) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
68 | 62, 67 | sylan9bbr 503 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ ¬ 𝑤 = 𝑧) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
69 | 58, 68 | sylan2 583 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ 𝑤 ∈ 𝑧) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
70 | 69 | adantrr 704 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ (𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢)) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
71 | 70 | pm5.32da 571 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ {𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢}) ↔ ((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))) |
72 | 23 | preleq 8875 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ {𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢}) → (𝑤 = (𝑓‘𝑢) ∧ 𝑧 = 𝑢)) |
73 | 71, 72 | syl6bir 246 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → (𝑤 = (𝑓‘𝑢) ∧ 𝑧 = 𝑢))) |
74 | 51 | eqeq2d 2788 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑧 = 𝑢 → (𝑤 = (𝑓‘𝑧) ↔ 𝑤 = (𝑓‘𝑢))) |
75 | 74 | biimparc 472 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑤 = (𝑓‘𝑢) ∧ 𝑧 = 𝑢) → 𝑤 = (𝑓‘𝑧)) |
76 | 73, 75 | syl6 35 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧))) |
77 | 76 | exp4c 425 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑤 ∈ 𝑧 → ((𝑓‘𝑢) ∈ 𝑢 → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧))))) |
78 | 77 | com13 88 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓‘𝑢) ∈ 𝑢 → (𝑤 ∈ 𝑧 → (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧))))) |
79 | 56, 78 | syl8 76 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑢 ∈ 𝑥 → (𝑢 ≠ ∅ → (𝑤 ∈ 𝑧 → (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧))))))) |
80 | 79 | com4r 94 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑤 ∈ 𝑧 → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑢 ∈ 𝑥 → (𝑢 ≠ ∅ → (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧))))))) |
81 | 80 | imp 398 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑤 ∈ 𝑧 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → (𝑢 ∈ 𝑥 → (𝑢 ≠ ∅ → (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))))) |
82 | 81 | imp4a 415 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑤 ∈ 𝑧 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → (𝑢 ∈ 𝑥 → ((𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢}) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧))))) |
83 | 82 | com3l 89 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑢 ∈ 𝑥 → ((𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢}) → ((𝑤 ∈ 𝑧 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧))))) |
84 | 83 | rexlimiv 3225 |
. . . . . . . . . . . . . . . . . 18
⊢
(∃𝑢 ∈
𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢}) → ((𝑤 ∈ 𝑧 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))) |
85 | 49, 84 | sylbi 209 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → ((𝑤 ∈ 𝑧 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))) |
86 | 85 | expd 408 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → (𝑤 ∈ 𝑧 → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧))))) |
87 | 86 | com13 88 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑤 ∈ 𝑧 → (𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧))))) |
88 | 87 | imp4b 414 |
. . . . . . . . . . . . . 14
⊢
((∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ 𝑤 ∈ 𝑧) → ((𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧))) |
89 | 88 | exlimdv 1892 |
. . . . . . . . . . . . 13
⊢
((∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ 𝑤 ∈ 𝑧) → (∃𝑣(𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧))) |
90 | 44, 89 | syl5bi 234 |
. . . . . . . . . . . 12
⊢
((∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ 𝑤 ∈ 𝑧) → (∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧))) |
91 | 90 | expimpd 446 |
. . . . . . . . . . 11
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧))) |
92 | 91 | alrimiv 1886 |
. . . . . . . . . 10
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑤((𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧))) |
93 | | mo2icl 3621 |
. . . . . . . . . 10
⊢
(∀𝑤((𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧)) → ∃*𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
94 | 92, 93 | syl 17 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃*𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
95 | 43, 94 | jctird 519 |
. . . . . . . 8
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → (∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ∧ ∃*𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))) |
96 | | df-reu 3095 |
. . . . . . . . 9
⊢
(∃!𝑤 ∈
𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃!𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
97 | | df-eu 2583 |
. . . . . . . . 9
⊢
(∃!𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ↔ (∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ∧ ∃*𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))) |
98 | 96, 97 | bitri 267 |
. . . . . . . 8
⊢
(∃!𝑤 ∈
𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ∧ ∃*𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))) |
99 | 95, 98 | syl6ibr 244 |
. . . . . . 7
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
100 | 99 | expd 408 |
. . . . . 6
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))) |
101 | 2, 100 | ralrimi 3166 |
. . . . 5
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
102 | | vex 3418 |
. . . . . . . . . . 11
⊢ 𝑓 ∈ V |
103 | 102 | rnex 7434 |
. . . . . . . . . 10
⊢ ran 𝑓 ∈ V |
104 | | p0ex 5138 |
. . . . . . . . . 10
⊢ {∅}
∈ V |
105 | 103, 104 | unex 7288 |
. . . . . . . . 9
⊢ (ran
𝑓 ∪ {∅}) ∈
V |
106 | | vex 3418 |
. . . . . . . . 9
⊢ 𝑥 ∈ V |
107 | 105, 106 | unex 7288 |
. . . . . . . 8
⊢ ((ran
𝑓 ∪ {∅}) ∪
𝑥) ∈
V |
108 | 107 | pwex 5135 |
. . . . . . 7
⊢ 𝒫
((ran 𝑓 ∪ {∅})
∪ 𝑥) ∈
V |
109 | | ssun1 4039 |
. . . . . . . . . . . . . 14
⊢ (ran
𝑓 ∪ {∅}) ⊆
((ran 𝑓 ∪ {∅})
∪ 𝑥) |
110 | | fvrn0 6529 |
. . . . . . . . . . . . . 14
⊢ (𝑓‘𝑢) ∈ (ran 𝑓 ∪ {∅}) |
111 | 109, 110 | sselii 3857 |
. . . . . . . . . . . . 13
⊢ (𝑓‘𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥) |
112 | | elun2 4044 |
. . . . . . . . . . . . 13
⊢ (𝑢 ∈ 𝑥 → 𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) |
113 | | prssi 4629 |
. . . . . . . . . . . . 13
⊢ (((𝑓‘𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∧ 𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) → {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) |
114 | 111, 112,
113 | sylancr 578 |
. . . . . . . . . . . 12
⊢ (𝑢 ∈ 𝑥 → {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) |
115 | | prex 5190 |
. . . . . . . . . . . . 13
⊢ {(𝑓‘𝑢), 𝑢} ∈ V |
116 | 115 | elpw 4429 |
. . . . . . . . . . . 12
⊢ ({(𝑓‘𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ↔ {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) |
117 | 114, 116 | sylibr 226 |
. . . . . . . . . . 11
⊢ (𝑢 ∈ 𝑥 → {(𝑓‘𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) |
118 | | eleq1 2853 |
. . . . . . . . . . 11
⊢ (𝑔 = {(𝑓‘𝑢), 𝑢} → (𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ↔ {(𝑓‘𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))) |
119 | 117, 118 | syl5ibrcom 239 |
. . . . . . . . . 10
⊢ (𝑢 ∈ 𝑥 → (𝑔 = {(𝑓‘𝑢), 𝑢} → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))) |
120 | 119 | adantld 483 |
. . . . . . . . 9
⊢ (𝑢 ∈ 𝑥 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))) |
121 | 120 | rexlimiv 3225 |
. . . . . . . 8
⊢
(∃𝑢 ∈
𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) |
122 | 121 | abssi 3938 |
. . . . . . 7
⊢ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ⊆ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) |
123 | 108, 122 | ssexi 5083 |
. . . . . 6
⊢ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∈ V |
124 | | rexeq 3346 |
. . . . . . . . 9
⊢ (𝑦 = {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → (∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
125 | 124 | reubidv 3329 |
. . . . . . . 8
⊢ (𝑦 = {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
126 | 125 | imbi2d 333 |
. . . . . . 7
⊢ (𝑦 = {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → ((𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ↔ (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))) |
127 | 126 | ralbidv 3147 |
. . . . . 6
⊢ (𝑦 = {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))) |
128 | 123, 127 | spcev 3525 |
. . . . 5
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
129 | 101, 128 | syl 17 |
. . . 4
⊢
(∀𝑧 ∈
𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
130 | 129 | exlimiv 1889 |
. . 3
⊢
(∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
131 | 130 | alimi 1774 |
. 2
⊢
(∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |
132 | 1, 131 | sylbi 209 |
1
⊢
(CHOICE → ∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) |