| Step | Hyp | Ref
| Expression |
| 1 | | df-ne 2941 |
. . . . . . . . 9
⊢ ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ ↔ ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) |
| 2 | 1 | ralbii 3093 |
. . . . . . . 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 3086 |
. . . . . . . . 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 217 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑥 ∈ On (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) |
| 6 | 2, 5 | biimtrrid 243 |
. . . . . . 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 8484 |
. . . . . . . 8
⊢
(∀𝑥 ∈ On
(𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V) |
| 9 | | elex 3501 |
. . . . . . . 8
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) |
| 10 | 8, 9 | nsyl3 138 |
. . . . . . 7
⊢ (𝐴 ∈ 𝐵 → ¬ ∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) |
| 11 | 6, 10 | nsyli 157 |
. . . . . 6
⊢ (𝜑 → (𝐴 ∈ 𝐵 → ¬ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅)) |
| 12 | | dfrex2 3073 |
. . . . . 6
⊢
(∃𝑥 ∈ On
(𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ↔ ¬ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) |
| 13 | 11, 12 | imbitrrdi 252 |
. . . . 5
⊢ (𝜑 → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On (𝐴 ∖ (𝐹 “ 𝑥)) = ∅)) |
| 14 | | imaeq2 6074 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦)) |
| 15 | 14 | difeq2d 4126 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝐴 ∖ (𝐹 “ 𝑥)) = (𝐴 ∖ (𝐹 “ 𝑦))) |
| 16 | 15 | eqeq1d 2739 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅)) |
| 17 | 16 | onminex 7822 |
. . . . 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 2941 |
. . . . . . 7
⊢ ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ↔ ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅) |
| 20 | 19 | ralbii 3093 |
. . . . . 6
⊢
(∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ↔ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅) |
| 21 | 20 | anbi2i 623 |
. . . . 5
⊢ (((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ↔ ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅)) |
| 22 | 21 | rexbii 3094 |
. . . 4
⊢
(∃𝑥 ∈ On
((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ↔ ∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅)) |
| 23 | 18, 22 | imbitrrdi 252 |
. . 3
⊢ (𝜑 → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅))) |
| 24 | | nfra1 3284 |
. . . . 5
⊢
Ⅎ𝑥∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) |
| 25 | 3, 24 | nfxfr 1853 |
. . . 4
⊢
Ⅎ𝑥𝜑 |
| 26 | | simpllr 776 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) |
| 27 | | fnfun 6668 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐹 Fn On → Fun 𝐹) |
| 28 | 7, 27 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ Fun 𝐹 |
| 29 | | fvelima 6974 |
. . . . . . . . . . . . . . . 16
⊢ ((Fun
𝐹 ∧ 𝑧 ∈ (𝐹 “ 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑧) |
| 30 | 28, 29 | mpan 690 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ (𝐹 “ 𝑥) → ∃𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑧) |
| 31 | | nfv 1914 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑦𝜑 |
| 32 | | nfra1 3284 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑦∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ |
| 33 | 31, 32 | nfan 1899 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑦(𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) |
| 34 | | nfv 1914 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑦(𝑥 ∈ On → 𝑧 ∈ 𝐴) |
| 35 | | rsp 3247 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝑦 ∈ 𝑥 → (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅)) |
| 36 | 35 | adantld 490 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅)) |
| 37 | | onelon 6409 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) |
| 38 | 15 | neeq1d 3000 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 = 𝑦 → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅)) |
| 39 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) |
| 40 | 39, 15 | eleq12d 2835 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) ↔ (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))) |
| 41 | 38, 40 | imbi12d 344 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 = 𝑦 → (((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) ↔ ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
| 42 | 41 | rspcv 3618 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ On → (∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
| 43 | 3, 42 | biimtrid 242 |
. . . . . . . . . . . . . . . . . . . . . . . 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 406 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))) |
| 49 | 48 | expcomd 416 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑦 ∈ 𝑥 → (𝑥 ∈ On → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
| 50 | | eldifi 4131 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)) → (𝐹‘𝑦) ∈ 𝐴) |
| 51 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹‘𝑦) = 𝑧 → ((𝐹‘𝑦) ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) |
| 52 | 50, 51 | syl5ibcom 245 |
. . . . . . . . . . . . . . . . . 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 3266 |
. . . . . . . . . . . . . . 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 406 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → (𝑧 ∈ (𝐹 “ 𝑥) → 𝑧 ∈ 𝐴)) |
| 59 | 58 | ssrdv 3989 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → (𝐹 “ 𝑥) ⊆ 𝐴) |
| 60 | | ssdif0 4366 |
. . . . . . . . . . . 12
⊢ (𝐴 ⊆ (𝐹 “ 𝑥) ↔ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) |
| 61 | 60 | biimpri 228 |
. . . . . . . . . . 11
⊢ ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → 𝐴 ⊆ (𝐹 “ 𝑥)) |
| 62 | 59, 61 | anim12i 613 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → ((𝐹 “ 𝑥) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐹 “ 𝑥))) |
| 63 | | eqss 3999 |
. . . . . . . . . 10
⊢ ((𝐹 “ 𝑥) = 𝐴 ↔ ((𝐹 “ 𝑥) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐹 “ 𝑥))) |
| 64 | 62, 63 | sylibr 234 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → (𝐹 “ 𝑥) = 𝐴) |
| 65 | | onss 7805 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ On → 𝑥 ⊆ On) |
| 66 | 32, 31 | nfan 1899 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑦(∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) |
| 67 | | nfv 1914 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑦 𝑥 ⊆ On |
| 68 | 66, 67 | nfan 1899 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑦((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) |
| 69 | | nfv 1914 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑧(((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦 ∈ 𝑥) |
| 70 | | ssel 3977 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On)) |
| 71 | | onss 7805 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ On → 𝑦 ⊆ On) |
| 72 | 7 | fndmi 6672 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ dom 𝐹 = On |
| 73 | 71, 72 | sseqtrrdi 4025 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ On → 𝑦 ⊆ dom 𝐹) |
| 74 | | funfvima2 7251 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((Fun
𝐹 ∧ 𝑦 ⊆ dom 𝐹) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹 “ 𝑦))) |
| 75 | 28, 73, 74 | sylancr 587 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ On → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹 “ 𝑦))) |
| 76 | 70, 75 | syl6 35 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹 “ 𝑦)))) |
| 77 | 35 | com12 32 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅)) |
| 78 | 77 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅))) |
| 79 | 70, 78, 44 | syl10 79 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝜑 → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))))) |
| 80 | 79 | imp4a 422 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))) |
| 81 | | eldifn 4132 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)) → ¬ (𝐹‘𝑦) ∈ (𝐹 “ 𝑦)) |
| 82 | | eleq1a 2836 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → ((𝐹‘𝑦) = (𝐹‘𝑧) → (𝐹‘𝑦) ∈ (𝐹 “ 𝑦))) |
| 83 | 82 | con3d 152 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → (¬ (𝐹‘𝑦) ∈ (𝐹 “ 𝑦) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))) |
| 84 | 81, 83 | syl5com 31 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))) |
| 85 | 80, 84 | syl8 76 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))))) |
| 86 | 85 | com34 91 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))))) |
| 87 | 76, 86 | syldd 72 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑦 → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))))) |
| 88 | 87 | com4r 94 |
. . . . . . . . . . . . . . . . . . 19
⊢
((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))))) |
| 89 | 88 | imp31 417 |
. . . . . . . . . . . . . . . . . 18
⊢
((((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦 ∈ 𝑥) → (𝑧 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑧))) |
| 90 | 69, 89 | ralrimi 3257 |
. . . . . . . . . . . . . . . . 17
⊢
((((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦 ∈ 𝑥) → ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧)) |
| 91 | 90 | ex 412 |
. . . . . . . . . . . . . . . 16
⊢
(((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) → (𝑦 ∈ 𝑥 → ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧))) |
| 92 | 68, 91 | ralrimi 3257 |
. . . . . . . . . . . . . . 15
⊢
(((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) → ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧)) |
| 93 | 92 | ex 412 |
. . . . . . . . . . . . . 14
⊢
((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧))) |
| 94 | 93 | ancld 550 |
. . . . . . . . . . . . 13
⊢
((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧)))) |
| 95 | 7 | tz7.48lem 8481 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧)) → Fun ◡(𝐹 ↾ 𝑥)) |
| 96 | 65, 94, 95 | syl56 36 |
. . . . . . . . . . . 12
⊢
((∀𝑦 ∈
𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ∈ On → Fun ◡(𝐹 ↾ 𝑥))) |
| 97 | 96 | ancoms 458 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑥 ∈ On → Fun ◡(𝐹 ↾ 𝑥))) |
| 98 | 97 | imp 406 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → Fun ◡(𝐹 ↾ 𝑥)) |
| 99 | 98 | adantr 480 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → Fun ◡(𝐹 ↾ 𝑥)) |
| 100 | 26, 64, 99 | 3jca 1129 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥))) |
| 101 | 100 | exp41 434 |
. . . . . . 7
⊢ (𝜑 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝑥 ∈ On → ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))))) |
| 102 | 101 | 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)))))) |
| 103 | 102 | 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)))))) |
| 104 | 103 | imp4a 422 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ On → (((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥))))) |
| 105 | 25, 104 | reximdai 3261 |
. . 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)))) |
| 106 | 23, 105 | syld 47 |
. 2
⊢ (𝜑 → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))) |
| 107 | 106 | impcom 407 |
1
⊢ ((𝐴 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥))) |