| Step | Hyp | Ref
| Expression |
| 1 | | simpl 482 |
. . . 4
⊢ ((𝐵 ∈ dom
𝑅1 ∧ 𝐴 ∈ 𝐵) → 𝐵 ∈ dom
𝑅1) |
| 2 | | r1funlim 9806 |
. . . . . . . 8
⊢ (Fun
𝑅1 ∧ Lim dom 𝑅1) |
| 3 | 2 | simpri 485 |
. . . . . . 7
⊢ Lim dom
𝑅1 |
| 4 | | limord 6444 |
. . . . . . 7
⊢ (Lim dom
𝑅1 → Ord dom 𝑅1) |
| 5 | 3, 4 | ax-mp 5 |
. . . . . 6
⊢ Ord dom
𝑅1 |
| 6 | | ordsson 7803 |
. . . . . 6
⊢ (Ord dom
𝑅1 → dom 𝑅1 ⊆
On) |
| 7 | 5, 6 | ax-mp 5 |
. . . . 5
⊢ dom
𝑅1 ⊆ On |
| 8 | 7 | sseli 3979 |
. . . 4
⊢ (𝐵 ∈ dom
𝑅1 → 𝐵 ∈ On) |
| 9 | 1, 8 | syl 17 |
. . 3
⊢ ((𝐵 ∈ dom
𝑅1 ∧ 𝐴 ∈ 𝐵) → 𝐵 ∈ On) |
| 10 | | onelon 6409 |
. . . . 5
⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On) |
| 11 | 8, 10 | sylan 580 |
. . . 4
⊢ ((𝐵 ∈ dom
𝑅1 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On) |
| 12 | | onsuc 7831 |
. . . 4
⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) |
| 13 | 11, 12 | syl 17 |
. . 3
⊢ ((𝐵 ∈ dom
𝑅1 ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ∈ On) |
| 14 | | eloni 6394 |
. . . . . 6
⊢ (𝐵 ∈ On → Ord 𝐵) |
| 15 | | ordsucss 7838 |
. . . . . 6
⊢ (Ord
𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
| 16 | 14, 15 | syl 17 |
. . . . 5
⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
| 17 | 16 | imp 406 |
. . . 4
⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ 𝐵) |
| 18 | 8, 17 | sylan 580 |
. . 3
⊢ ((𝐵 ∈ dom
𝑅1 ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ 𝐵) |
| 19 | | eleq1 2829 |
. . . . . 6
⊢ (𝑥 = suc 𝐴 → (𝑥 ∈ dom 𝑅1 ↔ suc
𝐴 ∈ dom
𝑅1)) |
| 20 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑥 = suc 𝐴 → (𝑅1‘𝑥) =
(𝑅1‘suc 𝐴)) |
| 21 | 20 | eleq2d 2827 |
. . . . . 6
⊢ (𝑥 = suc 𝐴 → ((𝑅1‘𝐴) ∈
(𝑅1‘𝑥) ↔ (𝑅1‘𝐴) ∈
(𝑅1‘suc 𝐴))) |
| 22 | 19, 21 | imbi12d 344 |
. . . . 5
⊢ (𝑥 = suc 𝐴 → ((𝑥 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) ↔ (suc 𝐴 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘suc
𝐴)))) |
| 23 | | eleq1 2829 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅1 ↔
𝑦 ∈ dom
𝑅1)) |
| 24 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) =
(𝑅1‘𝑦)) |
| 25 | 24 | eleq2d 2827 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((𝑅1‘𝐴) ∈
(𝑅1‘𝑥) ↔ (𝑅1‘𝐴) ∈
(𝑅1‘𝑦))) |
| 26 | 23, 25 | imbi12d 344 |
. . . . 5
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) ↔ (𝑦 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘𝑦)))) |
| 27 | | eleq1 2829 |
. . . . . 6
⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ dom 𝑅1 ↔ suc
𝑦 ∈ dom
𝑅1)) |
| 28 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) =
(𝑅1‘suc 𝑦)) |
| 29 | 28 | eleq2d 2827 |
. . . . . 6
⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝐴) ∈
(𝑅1‘𝑥) ↔ (𝑅1‘𝐴) ∈
(𝑅1‘suc 𝑦))) |
| 30 | 27, 29 | imbi12d 344 |
. . . . 5
⊢ (𝑥 = suc 𝑦 → ((𝑥 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) ↔ (suc 𝑦 ∈ dom
𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc
𝑦)))) |
| 31 | | eleq1 2829 |
. . . . . 6
⊢ (𝑥 = 𝐵 → (𝑥 ∈ dom 𝑅1 ↔
𝐵 ∈ dom
𝑅1)) |
| 32 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑥 = 𝐵 → (𝑅1‘𝑥) =
(𝑅1‘𝐵)) |
| 33 | 32 | eleq2d 2827 |
. . . . . 6
⊢ (𝑥 = 𝐵 → ((𝑅1‘𝐴) ∈
(𝑅1‘𝑥) ↔ (𝑅1‘𝐴) ∈
(𝑅1‘𝐵))) |
| 34 | 31, 33 | imbi12d 344 |
. . . . 5
⊢ (𝑥 = 𝐵 → ((𝑥 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) ↔ (𝐵 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘𝐵)))) |
| 35 | | fvex 6919 |
. . . . . . . 8
⊢
(𝑅1‘𝐴) ∈ V |
| 36 | 35 | pwid 4622 |
. . . . . . 7
⊢
(𝑅1‘𝐴) ∈ 𝒫
(𝑅1‘𝐴) |
| 37 | | limsuc 7870 |
. . . . . . . . 9
⊢ (Lim dom
𝑅1 → (𝐴 ∈ dom 𝑅1 ↔ suc
𝐴 ∈ dom
𝑅1)) |
| 38 | 3, 37 | ax-mp 5 |
. . . . . . . 8
⊢ (𝐴 ∈ dom
𝑅1 ↔ suc 𝐴 ∈ dom
𝑅1) |
| 39 | | r1sucg 9809 |
. . . . . . . 8
⊢ (𝐴 ∈ dom
𝑅1 → (𝑅1‘suc 𝐴) = 𝒫
(𝑅1‘𝐴)) |
| 40 | 38, 39 | sylbir 235 |
. . . . . . 7
⊢ (suc
𝐴 ∈ dom
𝑅1 → (𝑅1‘suc 𝐴) = 𝒫
(𝑅1‘𝐴)) |
| 41 | 36, 40 | eleqtrrid 2848 |
. . . . . 6
⊢ (suc
𝐴 ∈ dom
𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc
𝐴)) |
| 42 | 41 | a1i 11 |
. . . . 5
⊢ (suc
𝐴 ∈ On → (suc
𝐴 ∈ dom
𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc
𝐴))) |
| 43 | | limsuc 7870 |
. . . . . . . 8
⊢ (Lim dom
𝑅1 → (𝑦 ∈ dom 𝑅1 ↔ suc
𝑦 ∈ dom
𝑅1)) |
| 44 | 3, 43 | ax-mp 5 |
. . . . . . 7
⊢ (𝑦 ∈ dom
𝑅1 ↔ suc 𝑦 ∈ dom
𝑅1) |
| 45 | | r1tr 9816 |
. . . . . . . . . . 11
⊢ Tr
(𝑅1‘𝑦) |
| 46 | | dftr4 5266 |
. . . . . . . . . . 11
⊢ (Tr
(𝑅1‘𝑦) ↔ (𝑅1‘𝑦) ⊆ 𝒫
(𝑅1‘𝑦)) |
| 47 | 45, 46 | mpbi 230 |
. . . . . . . . . 10
⊢
(𝑅1‘𝑦) ⊆ 𝒫
(𝑅1‘𝑦) |
| 48 | | r1sucg 9809 |
. . . . . . . . . 10
⊢ (𝑦 ∈ dom
𝑅1 → (𝑅1‘suc 𝑦) = 𝒫
(𝑅1‘𝑦)) |
| 49 | 47, 48 | sseqtrrid 4027 |
. . . . . . . . 9
⊢ (𝑦 ∈ dom
𝑅1 → (𝑅1‘𝑦) ⊆ (𝑅1‘suc
𝑦)) |
| 50 | 49 | sseld 3982 |
. . . . . . . 8
⊢ (𝑦 ∈ dom
𝑅1 → ((𝑅1‘𝐴) ∈ (𝑅1‘𝑦) →
(𝑅1‘𝐴) ∈ (𝑅1‘suc
𝑦))) |
| 51 | 50 | a2i 14 |
. . . . . . 7
⊢ ((𝑦 ∈ dom
𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑦)) → (𝑦 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘suc
𝑦))) |
| 52 | 44, 51 | biimtrrid 243 |
. . . . . 6
⊢ ((𝑦 ∈ dom
𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘𝑦)) → (suc 𝑦 ∈ dom
𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc
𝑦))) |
| 53 | 52 | a1i 11 |
. . . . 5
⊢ (((𝑦 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑦) → ((𝑦 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘𝑦)) → (suc 𝑦 ∈ dom
𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc
𝑦)))) |
| 54 | | simprl 771 |
. . . . . . . . . . . 12
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
suc 𝐴 ⊆ 𝑥) |
| 55 | | simplr 769 |
. . . . . . . . . . . . . 14
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
suc 𝐴 ∈
On) |
| 56 | | onsucb 7837 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
| 57 | 55, 56 | sylibr 234 |
. . . . . . . . . . . . 13
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
𝐴 ∈
On) |
| 58 | | limord 6444 |
. . . . . . . . . . . . . 14
⊢ (Lim
𝑥 → Ord 𝑥) |
| 59 | 58 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
Ord 𝑥) |
| 60 | | ordelsuc 7840 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥)) |
| 61 | 57, 59, 60 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
(𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥)) |
| 62 | 54, 61 | mpbird 257 |
. . . . . . . . . . 11
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
𝐴 ∈ 𝑥) |
| 63 | | limsuc 7870 |
. . . . . . . . . . . 12
⊢ (Lim
𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥)) |
| 64 | 63 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
(𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥)) |
| 65 | 62, 64 | mpbid 232 |
. . . . . . . . . 10
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
suc 𝐴 ∈ 𝑥) |
| 66 | | simprr 773 |
. . . . . . . . . . . . 13
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
𝑥 ∈ dom
𝑅1) |
| 67 | | ordtr1 6427 |
. . . . . . . . . . . . . 14
⊢ (Ord dom
𝑅1 → ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1) →
𝐴 ∈ dom
𝑅1)) |
| 68 | 5, 67 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1) →
𝐴 ∈ dom
𝑅1) |
| 69 | 62, 66, 68 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
𝐴 ∈ dom
𝑅1) |
| 70 | 69, 39 | syl 17 |
. . . . . . . . . . 11
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
(𝑅1‘suc 𝐴) = 𝒫
(𝑅1‘𝐴)) |
| 71 | 36, 70 | eleqtrrid 2848 |
. . . . . . . . . 10
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
(𝑅1‘𝐴) ∈ (𝑅1‘suc
𝐴)) |
| 72 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑦 = suc 𝐴 → (𝑅1‘𝑦) =
(𝑅1‘suc 𝐴)) |
| 73 | 72 | eleq2d 2827 |
. . . . . . . . . . 11
⊢ (𝑦 = suc 𝐴 → ((𝑅1‘𝐴) ∈
(𝑅1‘𝑦) ↔ (𝑅1‘𝐴) ∈
(𝑅1‘suc 𝐴))) |
| 74 | 73 | rspcev 3622 |
. . . . . . . . . 10
⊢ ((suc
𝐴 ∈ 𝑥 ∧ (𝑅1‘𝐴) ∈
(𝑅1‘suc 𝐴)) → ∃𝑦 ∈ 𝑥 (𝑅1‘𝐴) ∈
(𝑅1‘𝑦)) |
| 75 | 65, 71, 74 | syl2anc 584 |
. . . . . . . . 9
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
∃𝑦 ∈ 𝑥
(𝑅1‘𝐴) ∈ (𝑅1‘𝑦)) |
| 76 | | eliun 4995 |
. . . . . . . . 9
⊢
((𝑅1‘𝐴) ∈ ∪
𝑦 ∈ 𝑥 (𝑅1‘𝑦) ↔ ∃𝑦 ∈ 𝑥 (𝑅1‘𝐴) ∈
(𝑅1‘𝑦)) |
| 77 | 75, 76 | sylibr 234 |
. . . . . . . 8
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
(𝑅1‘𝐴) ∈ ∪
𝑦 ∈ 𝑥 (𝑅1‘𝑦)) |
| 78 | | simpll 767 |
. . . . . . . . 9
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
Lim 𝑥) |
| 79 | | r1limg 9811 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom
𝑅1 ∧ Lim 𝑥) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦)) |
| 80 | 66, 78, 79 | syl2anc 584 |
. . . . . . . 8
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
(𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦)) |
| 81 | 77, 80 | eleqtrrd 2844 |
. . . . . . 7
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1)) →
(𝑅1‘𝐴) ∈ (𝑅1‘𝑥)) |
| 82 | 81 | expr 456 |
. . . . . 6
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → (𝑥 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘𝑥))) |
| 83 | 82 | a1d 25 |
. . . . 5
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (suc 𝐴 ⊆ 𝑦 → (𝑦 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘𝑦))) → (𝑥 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘𝑥)))) |
| 84 | 22, 26, 30, 34, 42, 53, 83 | tfindsg 7882 |
. . . 4
⊢ (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝐵) → (𝐵 ∈ dom 𝑅1 →
(𝑅1‘𝐴) ∈ (𝑅1‘𝐵))) |
| 85 | 84 | impr 454 |
. . 3
⊢ (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ (suc 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ dom 𝑅1)) →
(𝑅1‘𝐴) ∈ (𝑅1‘𝐵)) |
| 86 | 9, 13, 18, 1, 85 | syl22anc 839 |
. 2
⊢ ((𝐵 ∈ dom
𝑅1 ∧ 𝐴 ∈ 𝐵) → (𝑅1‘𝐴) ∈
(𝑅1‘𝐵)) |
| 87 | 86 | ex 412 |
1
⊢ (𝐵 ∈ dom
𝑅1 → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈
(𝑅1‘𝐵))) |