| Step | Hyp | Ref
| Expression |
| 1 | | simpr 484 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) →
(𝑅1‘𝐴) ∈ WUni) |
| 2 | 1 | wun0 10758 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ∅
∈ (𝑅1‘𝐴)) |
| 3 | | elfvdm 6943 |
. . . . . 6
⊢ (∅
∈ (𝑅1‘𝐴) → 𝐴 ∈ dom
𝑅1) |
| 4 | 2, 3 | syl 17 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ dom
𝑅1) |
| 5 | | r1fnon 9807 |
. . . . . 6
⊢
𝑅1 Fn On |
| 6 | 5 | fndmi 6672 |
. . . . 5
⊢ dom
𝑅1 = On |
| 7 | 4, 6 | eleqtrdi 2851 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ On) |
| 8 | | eloni 6394 |
. . . 4
⊢ (𝐴 ∈ On → Ord 𝐴) |
| 9 | 7, 8 | syl 17 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → Ord 𝐴) |
| 10 | | n0i 4340 |
. . . . . 6
⊢ (∅
∈ (𝑅1‘𝐴) → ¬
(𝑅1‘𝐴) = ∅) |
| 11 | 2, 10 | syl 17 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬
(𝑅1‘𝐴) = ∅) |
| 12 | | fveq2 6906 |
. . . . . 6
⊢ (𝐴 = ∅ →
(𝑅1‘𝐴) =
(𝑅1‘∅)) |
| 13 | | r10 9808 |
. . . . . 6
⊢
(𝑅1‘∅) = ∅ |
| 14 | 12, 13 | eqtrdi 2793 |
. . . . 5
⊢ (𝐴 = ∅ →
(𝑅1‘𝐴) = ∅) |
| 15 | 11, 14 | nsyl 140 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬
𝐴 =
∅) |
| 16 | | onsuc 7831 |
. . . . . . . 8
⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) |
| 17 | 7, 16 | syl 17 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → suc 𝐴 ∈ On) |
| 18 | | sucidg 6465 |
. . . . . . . 8
⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴) |
| 19 | 7, 18 | syl 17 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → 𝐴 ∈ suc 𝐴) |
| 20 | | r1ord 9820 |
. . . . . . 7
⊢ (suc
𝐴 ∈ On → (𝐴 ∈ suc 𝐴 → (𝑅1‘𝐴) ∈
(𝑅1‘suc 𝐴))) |
| 21 | 17, 19, 20 | sylc 65 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) →
(𝑅1‘𝐴) ∈ (𝑅1‘suc
𝐴)) |
| 22 | | r1elwf 9836 |
. . . . . 6
⊢
((𝑅1‘𝐴) ∈ (𝑅1‘suc
𝐴) →
(𝑅1‘𝐴) ∈ ∪
(𝑅1 “ On)) |
| 23 | | wfelirr 9865 |
. . . . . 6
⊢
((𝑅1‘𝐴) ∈ ∪
(𝑅1 “ On) → ¬
(𝑅1‘𝐴) ∈ (𝑅1‘𝐴)) |
| 24 | 21, 22, 23 | 3syl 18 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬
(𝑅1‘𝐴) ∈ (𝑅1‘𝐴)) |
| 25 | | simprr 773 |
. . . . . . . . 9
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 = suc 𝑥) |
| 26 | 25 | fveq2d 6910 |
. . . . . . . 8
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) =
(𝑅1‘suc 𝑥)) |
| 27 | | r1suc 9810 |
. . . . . . . . 9
⊢ (𝑥 ∈ On →
(𝑅1‘suc 𝑥) = 𝒫
(𝑅1‘𝑥)) |
| 28 | 27 | ad2antrl 728 |
. . . . . . . 8
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘suc
𝑥) = 𝒫
(𝑅1‘𝑥)) |
| 29 | 26, 28 | eqtrd 2777 |
. . . . . . 7
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) = 𝒫
(𝑅1‘𝑥)) |
| 30 | | simplr 769 |
. . . . . . . 8
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) ∈ WUni) |
| 31 | 7 | adantr 480 |
. . . . . . . . 9
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 ∈ On) |
| 32 | | sucidg 6465 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ On → 𝑥 ∈ suc 𝑥) |
| 33 | 32 | ad2antrl 728 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ suc 𝑥) |
| 34 | 33, 25 | eleqtrrd 2844 |
. . . . . . . . 9
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ 𝐴) |
| 35 | | r1ord 9820 |
. . . . . . . . 9
⊢ (𝐴 ∈ On → (𝑥 ∈ 𝐴 → (𝑅1‘𝑥) ∈
(𝑅1‘𝐴))) |
| 36 | 31, 34, 35 | sylc 65 |
. . . . . . . 8
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝑥) ∈
(𝑅1‘𝐴)) |
| 37 | 30, 36 | wunpw 10747 |
. . . . . . 7
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝒫
(𝑅1‘𝑥) ∈ (𝑅1‘𝐴)) |
| 38 | 29, 37 | eqeltrd 2841 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘𝐴) ∈
(𝑅1‘𝐴)) |
| 39 | 38 | rexlimdvaa 3156 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) →
(∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝑅1‘𝐴) ∈
(𝑅1‘𝐴))) |
| 40 | 24, 39 | mtod 198 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬
∃𝑥 ∈ On 𝐴 = suc 𝑥) |
| 41 | | ioran 986 |
. . . 4
⊢ (¬
(𝐴 = ∅ ∨
∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) |
| 42 | 15, 40, 41 | sylanbrc 583 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → ¬
(𝐴 = ∅ ∨
∃𝑥 ∈ On 𝐴 = suc 𝑥)) |
| 43 | | dflim3 7868 |
. . 3
⊢ (Lim
𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥))) |
| 44 | 9, 42, 43 | sylanbrc 583 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅1‘𝐴) ∈ WUni) → Lim 𝐴) |
| 45 | | r1limwun 10776 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ∈ WUni) |
| 46 | 44, 45 | impbida 801 |
1
⊢ (𝐴 ∈ 𝑉 → ((𝑅1‘𝐴) ∈ WUni ↔ Lim 𝐴)) |