Step | Hyp | Ref
| Expression |
1 | | df-ne 2985 |
. . . . . . . . 9
⊢ ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ ↔ ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) |
2 | 1 | ralbii 3132 |
. . . . . . . 8
⊢
(∀𝑥 ∈ On
(𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ ↔ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) |
3 | | tz7.49.2 |
. . . . . . . . 9
⊢ (𝜑 ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) |
4 | | ralim 3129 |
. . . . . . . . 9
⊢
(∀𝑥 ∈ On
((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → (∀𝑥 ∈ On (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) |
5 | 3, 4 | sylbi 218 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑥 ∈ On (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) |
6 | 2, 5 | syl5bir 244 |
. . . . . . 7
⊢ (𝜑 → (∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) |
7 | | tz7.49.1 |
. . . . . . . . 9
⊢ 𝐹 Fn On |
8 | 7 | tz7.48-3 7936 |
. . . . . . . 8
⊢
(∀𝑥 ∈ On
(𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V) |
9 | | elex 3455 |
. . . . . . . 8
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) |
10 | 8, 9 | nsyl3 140 |
. . . . . . 7
⊢ (𝐴 ∈ 𝐵 → ¬ ∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) |
11 | 6, 10 | nsyli 160 |
. . . . . 6
⊢ (𝜑 → (𝐴 ∈ 𝐵 → ¬ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅)) |
12 | | dfrex2 3203 |
. . . . . 6
⊢
(∃𝑥 ∈ On
(𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ↔ ¬ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) |
13 | 11, 12 | syl6ibr 253 |
. . . . 5
⊢ (𝜑 → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On (𝐴 ∖ (𝐹 “ 𝑥)) = ∅)) |
14 | | imaeq2 5807 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦)) |
15 | 14 | difeq2d 4024 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝐴 ∖ (𝐹 “ 𝑥)) = (𝐴 ∖ (𝐹 “ 𝑦))) |
16 | 15 | eqeq1d 2797 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅)) |
17 | 16 | onminex 7383 |
. . . . 5
⊢
(∃𝑥 ∈ On
(𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅)) |
18 | 13, 17 | syl6 35 |
. . . 4
⊢ (𝜑 → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅))) |
19 | | df-ne 2985 |
. . . . . . 7
⊢ ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ↔ ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅) |
20 | 19 | ralbii 3132 |
. . . . . 6
⊢
(∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ↔ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅) |
21 | 20 | anbi2i 622 |
. . . . 5
⊢ (((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ↔ ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅)) |
22 | 21 | rexbii 3211 |
. . . 4
⊢
(∃𝑥 ∈ On
((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ↔ ∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅)) |
23 | 18, 22 | syl6ibr 253 |
. . 3
⊢ (𝜑 → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅))) |
24 | | nfra1 3186 |
. . . . 5
⊢
Ⅎ𝑥∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) |
25 | 3, 24 | nfxfr 1834 |
. . . 4
⊢
Ⅎ𝑥𝜑 |
26 | | simpllr 772 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) |
27 | | fnfun 6328 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐹 Fn On → Fun 𝐹) |
28 | 7, 27 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ Fun 𝐹 |
29 | | fvelima 6604 |
. . . . . . . . . . . . . . . 16
⊢ ((Fun
𝐹 ∧ 𝑧 ∈ (𝐹 “ 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑧) |
30 | 28, 29 | mpan 686 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ (𝐹 “ 𝑥) → ∃𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑧) |
31 | | nfv 1892 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑦𝜑 |
32 | | nfra1 3186 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑦∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ |
33 | 31, 32 | nfan 1881 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑦(𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) |
34 | | nfv 1892 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑦(𝑥 ∈ On → 𝑧 ∈ 𝐴) |
35 | | rsp 3172 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝑦 ∈ 𝑥 → (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅)) |
36 | 35 | adantld 491 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅)) |
37 | | onelon 6096 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) |
38 | 15 | neeq1d 3043 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 = 𝑦 → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅)) |
39 | | fveq2 6543 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) |
40 | 39, 15 | eleq12d 2877 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) ↔ (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))) |
41 | 38, 40 | imbi12d 346 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 = 𝑦 → (((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) ↔ ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
42 | 41 | rspcv 3555 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ On → (∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
43 | 3, 42 | syl5bi 243 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ On → (𝜑 → ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
44 | 43 | com23 86 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ On → ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝜑 → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
45 | 37, 44 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝜑 → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
46 | 36, 45 | sylcom 30 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝜑 → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
47 | 46 | com3r 87 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
48 | 47 | imp 407 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))) |
49 | 48 | expcomd 417 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑦 ∈ 𝑥 → (𝑥 ∈ On → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
50 | | eldifi 4028 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)) → (𝐹‘𝑦) ∈ 𝐴) |
51 | | eleq1 2870 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹‘𝑦) = 𝑧 → ((𝐹‘𝑦) ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) |
52 | 50, 51 | syl5ibcom 246 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)) → ((𝐹‘𝑦) = 𝑧 → 𝑧 ∈ 𝐴)) |
53 | 49, 52 | syl8 76 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑦 ∈ 𝑥 → (𝑥 ∈ On → ((𝐹‘𝑦) = 𝑧 → 𝑧 ∈ 𝐴)))) |
54 | 53 | com34 91 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑦 ∈ 𝑥 → ((𝐹‘𝑦) = 𝑧 → (𝑥 ∈ On → 𝑧 ∈ 𝐴)))) |
55 | 33, 34, 54 | rexlimd 3278 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (∃𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑧 → (𝑥 ∈ On → 𝑧 ∈ 𝐴))) |
56 | 30, 55 | syl5 34 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑧 ∈ (𝐹 “ 𝑥) → (𝑥 ∈ On → 𝑧 ∈ 𝐴))) |
57 | 56 | com23 86 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑥 ∈ On → (𝑧 ∈ (𝐹 “ 𝑥) → 𝑧 ∈ 𝐴))) |
58 | 57 | imp 407 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → (𝑧 ∈ (𝐹 “ 𝑥) → 𝑧 ∈ 𝐴)) |
59 | 58 | ssrdv 3899 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → (𝐹 “ 𝑥) ⊆ 𝐴) |
60 | | ssdif0 4247 |
. . . . . . . . . . . 12
⊢ (𝐴 ⊆ (𝐹 “ 𝑥) ↔ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) |
61 | 60 | biimpri 229 |
. . . . . . . . . . 11
⊢ ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → 𝐴 ⊆ (𝐹 “ 𝑥)) |
62 | 59, 61 | anim12i 612 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → ((𝐹 “ 𝑥) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐹 “ 𝑥))) |
63 | | eqss 3908 |
. . . . . . . . . 10
⊢ ((𝐹 “ 𝑥) = 𝐴 ↔ ((𝐹 “ 𝑥) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐹 “ 𝑥))) |
64 | 62, 63 | sylibr 235 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → (𝐹 “ 𝑥) = 𝐴) |
65 | | onss 7366 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ On → 𝑥 ⊆ On) |
66 | 32, 31 | nfan 1881 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑦(∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) |
67 | | nfv 1892 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑦 𝑥 ⊆ On |
68 | 66, 67 | nfan 1881 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑦((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) |
69 | | nfv 1892 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑧(((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦 ∈ 𝑥) |
70 | | ssel 3887 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On)) |
71 | | onss 7366 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ On → 𝑦 ⊆ On) |
72 | | fndm 6330 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝐹 Fn On → dom 𝐹 = On) |
73 | 7, 72 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ dom 𝐹 = On |
74 | 71, 73 | syl6sseqr 3943 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ On → 𝑦 ⊆ dom 𝐹) |
75 | | funfvima2 6864 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((Fun
𝐹 ∧ 𝑦 ⊆ dom 𝐹) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹 “ 𝑦))) |
76 | 28, 74, 75 | sylancr 587 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ On → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹 “ 𝑦))) |
77 | 70, 76 | syl6 35 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹 “ 𝑦)))) |
78 | 35 | com12 32 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅)) |
79 | 78 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅))) |
80 | 70, 79, 44 | syl10 79 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝜑 → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))))) |
81 | 80 | imp4a 423 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
82 | | eldifn 4029 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)) → ¬ (𝐹‘𝑦) ∈ (𝐹 “ 𝑦)) |
83 | | eleq1a 2878 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → ((𝐹‘𝑦) = (𝐹‘𝑧) → (𝐹‘𝑦) ∈ (𝐹 “ 𝑦))) |
84 | 83 | con3d 155 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → (¬ (𝐹‘𝑦) ∈ (𝐹 “ 𝑦) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))) |
85 | 82, 84 | syl5com 31 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))) |
86 | 81, 85 | syl8 76 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))))) |
87 | 86 | com34 91 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))))) |
88 | 77, 87 | syldd 72 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑦 → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))))) |
89 | 88 | com4r 94 |
. . . . . . . . . . . . . . . . . . 19
⊢
((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))))) |
90 | 89 | imp31 418 |
. . . . . . . . . . . . . . . . . 18
⊢
((((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦 ∈ 𝑥) → (𝑧 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))) |
91 | 69, 90 | ralrimi 3183 |
. . . . . . . . . . . . . . . . 17
⊢
((((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦 ∈ 𝑥) → ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧)) |
92 | 91 | ex 413 |
. . . . . . . . . . . . . . . 16
⊢
(((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) → (𝑦 ∈ 𝑥 → ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧))) |
93 | 68, 92 | ralrimi 3183 |
. . . . . . . . . . . . . . 15
⊢
(((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) → ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧)) |
94 | 93 | ex 413 |
. . . . . . . . . . . . . 14
⊢
((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧))) |
95 | 94 | ancld 551 |
. . . . . . . . . . . . 13
⊢
((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧)))) |
96 | 7 | tz7.48lem 7933 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧)) → Fun ◡(𝐹 ↾ 𝑥)) |
97 | 65, 95, 96 | syl56 36 |
. . . . . . . . . . . 12
⊢
((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ∈ On → Fun ◡(𝐹 ↾ 𝑥))) |
98 | 97 | ancoms 459 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑥 ∈ On → Fun ◡(𝐹 ↾ 𝑥))) |
99 | 98 | imp 407 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → Fun ◡(𝐹 ↾ 𝑥)) |
100 | 99 | adantr 481 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → Fun ◡(𝐹 ↾ 𝑥)) |
101 | 26, 64, 100 | 3jca 1121 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥))) |
102 | 101 | exp41 435 |
. . . . . . 7
⊢ (𝜑 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝑥 ∈ On → ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))))) |
103 | 102 | com23 86 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))))) |
104 | 103 | com34 91 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ On → ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))))) |
105 | 104 | imp4a 423 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ On → (((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥))))) |
106 | 25, 105 | reximdai 3272 |
. . 3
⊢ (𝜑 → (∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))) |
107 | 23, 106 | syld 47 |
. 2
⊢ (𝜑 → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))) |
108 | 107 | impcom 408 |
1
⊢ ((𝐴 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥))) |