| Step | Hyp | Ref
| Expression |
| 1 | | hsmexlem4.o |
. . . . . . 7
⊢ 𝑂 = OrdIso( E , (rank “
((𝑈‘𝑑)‘𝑐))) |
| 2 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑐 = ∅ → ((𝑈‘𝑑)‘𝑐) = ((𝑈‘𝑑)‘∅)) |
| 3 | 2 | imaeq2d 6078 |
. . . . . . . 8
⊢ (𝑐 = ∅ → (rank “
((𝑈‘𝑑)‘𝑐)) = (rank “ ((𝑈‘𝑑)‘∅))) |
| 4 | | oieq2 9553 |
. . . . . . . 8
⊢ ((rank
“ ((𝑈‘𝑑)‘𝑐)) = (rank “ ((𝑈‘𝑑)‘∅)) → OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈‘𝑑)‘∅)))) |
| 5 | 3, 4 | syl 17 |
. . . . . . 7
⊢ (𝑐 = ∅ → OrdIso( E ,
(rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈‘𝑑)‘∅)))) |
| 6 | 1, 5 | eqtrid 2789 |
. . . . . 6
⊢ (𝑐 = ∅ → 𝑂 = OrdIso( E , (rank “
((𝑈‘𝑑)‘∅)))) |
| 7 | 6 | dmeqd 5916 |
. . . . 5
⊢ (𝑐 = ∅ → dom 𝑂 = dom OrdIso( E , (rank “
((𝑈‘𝑑)‘∅)))) |
| 8 | | fveq2 6906 |
. . . . 5
⊢ (𝑐 = ∅ → (𝐻‘𝑐) = (𝐻‘∅)) |
| 9 | 7, 8 | eleq12d 2835 |
. . . 4
⊢ (𝑐 = ∅ → (dom 𝑂 ∈ (𝐻‘𝑐) ↔ dom OrdIso( E , (rank “
((𝑈‘𝑑)‘∅))) ∈ (𝐻‘∅))) |
| 10 | 9 | ralbidv 3178 |
. . 3
⊢ (𝑐 = ∅ → (∀𝑑 ∈ 𝑆 dom 𝑂 ∈ (𝐻‘𝑐) ↔ ∀𝑑 ∈ 𝑆 dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘∅))) ∈ (𝐻‘∅))) |
| 11 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑐 = 𝑒 → ((𝑈‘𝑑)‘𝑐) = ((𝑈‘𝑑)‘𝑒)) |
| 12 | 11 | imaeq2d 6078 |
. . . . . . . 8
⊢ (𝑐 = 𝑒 → (rank “ ((𝑈‘𝑑)‘𝑐)) = (rank “ ((𝑈‘𝑑)‘𝑒))) |
| 13 | | oieq2 9553 |
. . . . . . . 8
⊢ ((rank
“ ((𝑈‘𝑑)‘𝑐)) = (rank “ ((𝑈‘𝑑)‘𝑒)) → OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑒)))) |
| 14 | 12, 13 | syl 17 |
. . . . . . 7
⊢ (𝑐 = 𝑒 → OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑒)))) |
| 15 | 1, 14 | eqtrid 2789 |
. . . . . 6
⊢ (𝑐 = 𝑒 → 𝑂 = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑒)))) |
| 16 | 15 | dmeqd 5916 |
. . . . 5
⊢ (𝑐 = 𝑒 → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑒)))) |
| 17 | | fveq2 6906 |
. . . . 5
⊢ (𝑐 = 𝑒 → (𝐻‘𝑐) = (𝐻‘𝑒)) |
| 18 | 16, 17 | eleq12d 2835 |
. . . 4
⊢ (𝑐 = 𝑒 → (dom 𝑂 ∈ (𝐻‘𝑐) ↔ dom OrdIso( E , (rank “
((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒))) |
| 19 | 18 | ralbidv 3178 |
. . 3
⊢ (𝑐 = 𝑒 → (∀𝑑 ∈ 𝑆 dom 𝑂 ∈ (𝐻‘𝑐) ↔ ∀𝑑 ∈ 𝑆 dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒))) |
| 20 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑐 = suc 𝑒 → ((𝑈‘𝑑)‘𝑐) = ((𝑈‘𝑑)‘suc 𝑒)) |
| 21 | 20 | imaeq2d 6078 |
. . . . . . . 8
⊢ (𝑐 = suc 𝑒 → (rank “ ((𝑈‘𝑑)‘𝑐)) = (rank “ ((𝑈‘𝑑)‘suc 𝑒))) |
| 22 | | oieq2 9553 |
. . . . . . . 8
⊢ ((rank
“ ((𝑈‘𝑑)‘𝑐)) = (rank “ ((𝑈‘𝑑)‘suc 𝑒)) → OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈‘𝑑)‘suc 𝑒)))) |
| 23 | 21, 22 | syl 17 |
. . . . . . 7
⊢ (𝑐 = suc 𝑒 → OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈‘𝑑)‘suc 𝑒)))) |
| 24 | 1, 23 | eqtrid 2789 |
. . . . . 6
⊢ (𝑐 = suc 𝑒 → 𝑂 = OrdIso( E , (rank “ ((𝑈‘𝑑)‘suc 𝑒)))) |
| 25 | 24 | dmeqd 5916 |
. . . . 5
⊢ (𝑐 = suc 𝑒 → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘suc 𝑒)))) |
| 26 | | fveq2 6906 |
. . . . 5
⊢ (𝑐 = suc 𝑒 → (𝐻‘𝑐) = (𝐻‘suc 𝑒)) |
| 27 | 25, 26 | eleq12d 2835 |
. . . 4
⊢ (𝑐 = suc 𝑒 → (dom 𝑂 ∈ (𝐻‘𝑐) ↔ dom OrdIso( E , (rank “
((𝑈‘𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒))) |
| 28 | 27 | ralbidv 3178 |
. . 3
⊢ (𝑐 = suc 𝑒 → (∀𝑑 ∈ 𝑆 dom 𝑂 ∈ (𝐻‘𝑐) ↔ ∀𝑑 ∈ 𝑆 dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒))) |
| 29 | | imassrn 6089 |
. . . . . . 7
⊢ (rank
“ ((𝑈‘𝑑)‘∅)) ⊆ ran
rank |
| 30 | | rankf 9834 |
. . . . . . . 8
⊢
rank:∪ (𝑅1 “
On)⟶On |
| 31 | | frn 6743 |
. . . . . . . 8
⊢
(rank:∪ (𝑅1 “
On)⟶On → ran rank ⊆ On) |
| 32 | 30, 31 | ax-mp 5 |
. . . . . . 7
⊢ ran rank
⊆ On |
| 33 | 29, 32 | sstri 3993 |
. . . . . 6
⊢ (rank
“ ((𝑈‘𝑑)‘∅)) ⊆
On |
| 34 | | hsmexlem4.u |
. . . . . . . . . 10
⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦),
𝑥) ↾
ω)) |
| 35 | 34 | ituni0 10458 |
. . . . . . . . 9
⊢ (𝑑 ∈ V → ((𝑈‘𝑑)‘∅) = 𝑑) |
| 36 | 35 | elv 3485 |
. . . . . . . 8
⊢ ((𝑈‘𝑑)‘∅) = 𝑑 |
| 37 | 36 | imaeq2i 6076 |
. . . . . . 7
⊢ (rank
“ ((𝑈‘𝑑)‘∅)) = (rank
“ 𝑑) |
| 38 | | ffun 6739 |
. . . . . . . . . 10
⊢
(rank:∪ (𝑅1 “
On)⟶On → Fun rank) |
| 39 | 30, 38 | ax-mp 5 |
. . . . . . . . 9
⊢ Fun
rank |
| 40 | | vex 3484 |
. . . . . . . . 9
⊢ 𝑑 ∈ V |
| 41 | | wdomimag 9627 |
. . . . . . . . 9
⊢ ((Fun
rank ∧ 𝑑 ∈ V)
→ (rank “ 𝑑)
≼* 𝑑) |
| 42 | 39, 40, 41 | mp2an 692 |
. . . . . . . 8
⊢ (rank
“ 𝑑)
≼* 𝑑 |
| 43 | | sneq 4636 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑑 → {𝑎} = {𝑑}) |
| 44 | 43 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑑 → (TC‘{𝑎}) = (TC‘{𝑑})) |
| 45 | 44 | raleqdv 3326 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑑 → (∀𝑏 ∈ (TC‘{𝑎})𝑏 ≼ 𝑋 ↔ ∀𝑏 ∈ (TC‘{𝑑})𝑏 ≼ 𝑋)) |
| 46 | | hsmexlem4.s |
. . . . . . . . . . 11
⊢ 𝑆 = {𝑎 ∈ ∪
(𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏 ≼ 𝑋} |
| 47 | 45, 46 | elrab2 3695 |
. . . . . . . . . 10
⊢ (𝑑 ∈ 𝑆 ↔ (𝑑 ∈ ∪
(𝑅1 “ On) ∧ ∀𝑏 ∈ (TC‘{𝑑})𝑏 ≼ 𝑋)) |
| 48 | 47 | simprbi 496 |
. . . . . . . . 9
⊢ (𝑑 ∈ 𝑆 → ∀𝑏 ∈ (TC‘{𝑑})𝑏 ≼ 𝑋) |
| 49 | | vsnex 5434 |
. . . . . . . . . . . 12
⊢ {𝑑} ∈ V |
| 50 | | tcid 9779 |
. . . . . . . . . . . 12
⊢ ({𝑑} ∈ V → {𝑑} ⊆ (TC‘{𝑑})) |
| 51 | 49, 50 | ax-mp 5 |
. . . . . . . . . . 11
⊢ {𝑑} ⊆ (TC‘{𝑑}) |
| 52 | | vsnid 4663 |
. . . . . . . . . . 11
⊢ 𝑑 ∈ {𝑑} |
| 53 | 51, 52 | sselii 3980 |
. . . . . . . . . 10
⊢ 𝑑 ∈ (TC‘{𝑑}) |
| 54 | | breq1 5146 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑑 → (𝑏 ≼ 𝑋 ↔ 𝑑 ≼ 𝑋)) |
| 55 | 54 | rspcv 3618 |
. . . . . . . . . 10
⊢ (𝑑 ∈ (TC‘{𝑑}) → (∀𝑏 ∈ (TC‘{𝑑})𝑏 ≼ 𝑋 → 𝑑 ≼ 𝑋)) |
| 56 | 53, 55 | ax-mp 5 |
. . . . . . . . 9
⊢
(∀𝑏 ∈
(TC‘{𝑑})𝑏 ≼ 𝑋 → 𝑑 ≼ 𝑋) |
| 57 | | domwdom 9614 |
. . . . . . . . 9
⊢ (𝑑 ≼ 𝑋 → 𝑑 ≼* 𝑋) |
| 58 | 48, 56, 57 | 3syl 18 |
. . . . . . . 8
⊢ (𝑑 ∈ 𝑆 → 𝑑 ≼* 𝑋) |
| 59 | | wdomtr 9615 |
. . . . . . . 8
⊢ (((rank
“ 𝑑)
≼* 𝑑 ∧
𝑑 ≼* 𝑋) → (rank “ 𝑑) ≼* 𝑋) |
| 60 | 42, 58, 59 | sylancr 587 |
. . . . . . 7
⊢ (𝑑 ∈ 𝑆 → (rank “ 𝑑) ≼* 𝑋) |
| 61 | 37, 60 | eqbrtrid 5178 |
. . . . . 6
⊢ (𝑑 ∈ 𝑆 → (rank “ ((𝑈‘𝑑)‘∅)) ≼* 𝑋) |
| 62 | | eqid 2737 |
. . . . . . 7
⊢ OrdIso( E
, (rank “ ((𝑈‘𝑑)‘∅))) = OrdIso( E , (rank
“ ((𝑈‘𝑑)‘∅))) |
| 63 | 62 | hsmexlem1 10466 |
. . . . . 6
⊢ (((rank
“ ((𝑈‘𝑑)‘∅)) ⊆ On
∧ (rank “ ((𝑈‘𝑑)‘∅)) ≼* 𝑋) → dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘∅))) ∈
(har‘𝒫 𝑋)) |
| 64 | 33, 61, 63 | sylancr 587 |
. . . . 5
⊢ (𝑑 ∈ 𝑆 → dom OrdIso( E , (rank “
((𝑈‘𝑑)‘∅))) ∈
(har‘𝒫 𝑋)) |
| 65 | | hsmexlem4.h |
. . . . . 6
⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) |
| 66 | 65 | hsmexlem7 10463 |
. . . . 5
⊢ (𝐻‘∅) =
(har‘𝒫 𝑋) |
| 67 | 64, 66 | eleqtrrdi 2852 |
. . . 4
⊢ (𝑑 ∈ 𝑆 → dom OrdIso( E , (rank “
((𝑈‘𝑑)‘∅))) ∈ (𝐻‘∅)) |
| 68 | 67 | rgen 3063 |
. . 3
⊢
∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘∅))) ∈ (𝐻‘∅) |
| 69 | | nfra1 3284 |
. . . . . 6
⊢
Ⅎ𝑑∀𝑑 ∈ 𝑆 dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) |
| 70 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑑 𝑒 ∈ ω |
| 71 | 69, 70 | nfan 1899 |
. . . . 5
⊢
Ⅎ𝑑(∀𝑑 ∈ 𝑆 dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ 𝑒 ∈ ω) |
| 72 | 34 | ituniiun 10462 |
. . . . . . . . . . . . 13
⊢ (𝑑 ∈ V → ((𝑈‘𝑑)‘suc 𝑒) = ∪ 𝑓 ∈ 𝑑 ((𝑈‘𝑓)‘𝑒)) |
| 73 | 72 | elv 3485 |
. . . . . . . . . . . 12
⊢ ((𝑈‘𝑑)‘suc 𝑒) = ∪ 𝑓 ∈ 𝑑 ((𝑈‘𝑓)‘𝑒) |
| 74 | 73 | imaeq2i 6076 |
. . . . . . . . . . 11
⊢ (rank
“ ((𝑈‘𝑑)‘suc 𝑒)) = (rank “ ∪ 𝑓 ∈ 𝑑 ((𝑈‘𝑓)‘𝑒)) |
| 75 | | imaiun 7265 |
. . . . . . . . . . 11
⊢ (rank
“ ∪ 𝑓 ∈ 𝑑 ((𝑈‘𝑓)‘𝑒)) = ∪
𝑓 ∈ 𝑑 (rank “ ((𝑈‘𝑓)‘𝑒)) |
| 76 | 74, 75 | eqtri 2765 |
. . . . . . . . . 10
⊢ (rank
“ ((𝑈‘𝑑)‘suc 𝑒)) = ∪
𝑓 ∈ 𝑑 (rank “ ((𝑈‘𝑓)‘𝑒)) |
| 77 | | oieq2 9553 |
. . . . . . . . . 10
⊢ ((rank
“ ((𝑈‘𝑑)‘suc 𝑒)) = ∪
𝑓 ∈ 𝑑 (rank “ ((𝑈‘𝑓)‘𝑒)) → OrdIso( E , (rank “ ((𝑈‘𝑑)‘suc 𝑒))) = OrdIso( E , ∪ 𝑓 ∈ 𝑑 (rank “ ((𝑈‘𝑓)‘𝑒)))) |
| 78 | 76, 77 | ax-mp 5 |
. . . . . . . . 9
⊢ OrdIso( E
, (rank “ ((𝑈‘𝑑)‘suc 𝑒))) = OrdIso( E , ∪ 𝑓 ∈ 𝑑 (rank “ ((𝑈‘𝑓)‘𝑒))) |
| 79 | 78 | dmeqi 5915 |
. . . . . . . 8
⊢ dom
OrdIso( E , (rank “ ((𝑈‘𝑑)‘suc 𝑒))) = dom OrdIso( E , ∪ 𝑓 ∈ 𝑑 (rank “ ((𝑈‘𝑓)‘𝑒))) |
| 80 | 58 | ad2antll 729 |
. . . . . . . . 9
⊢
((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ 𝑆)) → 𝑑 ≼* 𝑋) |
| 81 | 65 | hsmexlem9 10465 |
. . . . . . . . . 10
⊢ (𝑒 ∈ ω → (𝐻‘𝑒) ∈ On) |
| 82 | 81 | ad2antrl 728 |
. . . . . . . . 9
⊢
((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ 𝑆)) → (𝐻‘𝑒) ∈ On) |
| 83 | | fveq2 6906 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑑 = 𝑓 → (𝑈‘𝑑) = (𝑈‘𝑓)) |
| 84 | 83 | fveq1d 6908 |
. . . . . . . . . . . . . . . 16
⊢ (𝑑 = 𝑓 → ((𝑈‘𝑑)‘𝑒) = ((𝑈‘𝑓)‘𝑒)) |
| 85 | 84 | imaeq2d 6078 |
. . . . . . . . . . . . . . 15
⊢ (𝑑 = 𝑓 → (rank “ ((𝑈‘𝑑)‘𝑒)) = (rank “ ((𝑈‘𝑓)‘𝑒))) |
| 86 | | oieq2 9553 |
. . . . . . . . . . . . . . 15
⊢ ((rank
“ ((𝑈‘𝑑)‘𝑒)) = (rank “ ((𝑈‘𝑓)‘𝑒)) → OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑒))) = OrdIso( E , (rank “ ((𝑈‘𝑓)‘𝑒)))) |
| 87 | 85, 86 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑑 = 𝑓 → OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑒))) = OrdIso( E , (rank “ ((𝑈‘𝑓)‘𝑒)))) |
| 88 | 87 | dmeqd 5916 |
. . . . . . . . . . . . 13
⊢ (𝑑 = 𝑓 → dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑒))) = dom OrdIso( E , (rank “ ((𝑈‘𝑓)‘𝑒)))) |
| 89 | 88 | eleq1d 2826 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑓 → (dom OrdIso( E , (rank “
((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ↔ dom OrdIso( E , (rank “
((𝑈‘𝑓)‘𝑒))) ∈ (𝐻‘𝑒))) |
| 90 | | simpll 767 |
. . . . . . . . . . . 12
⊢
(((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ 𝑆)) ∧ 𝑓 ∈ 𝑑) → ∀𝑑 ∈ 𝑆 dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒)) |
| 91 | 46 | ssrab3 4082 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑆 ⊆ ∪ (𝑅1 “ On) |
| 92 | 91 | sseli 3979 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪
(𝑅1 “ On)) |
| 93 | | r1elssi 9845 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑑 ∈ ∪ (𝑅1 “ On) → 𝑑 ⊆ ∪ (𝑅1 “ On)) |
| 94 | 92, 93 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑑 ∈ 𝑆 → 𝑑 ⊆ ∪
(𝑅1 “ On)) |
| 95 | 94 | sselda 3983 |
. . . . . . . . . . . . . . 15
⊢ ((𝑑 ∈ 𝑆 ∧ 𝑓 ∈ 𝑑) → 𝑓 ∈ ∪
(𝑅1 “ On)) |
| 96 | | snssi 4808 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓 ∈ 𝑑 → {𝑓} ⊆ 𝑑) |
| 97 | 40 | tcss 9784 |
. . . . . . . . . . . . . . . . . . 19
⊢ ({𝑓} ⊆ 𝑑 → (TC‘{𝑓}) ⊆ (TC‘𝑑)) |
| 98 | 96, 97 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓 ∈ 𝑑 → (TC‘{𝑓}) ⊆ (TC‘𝑑)) |
| 99 | 49 | tcel 9785 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ∈ {𝑑} → (TC‘𝑑) ⊆ (TC‘{𝑑})) |
| 100 | 52, 99 | mp1i 13 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓 ∈ 𝑑 → (TC‘𝑑) ⊆ (TC‘{𝑑})) |
| 101 | 98, 100 | sstrd 3994 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 ∈ 𝑑 → (TC‘{𝑓}) ⊆ (TC‘{𝑑})) |
| 102 | | ssralv 4052 |
. . . . . . . . . . . . . . . . 17
⊢
((TC‘{𝑓})
⊆ (TC‘{𝑑})
→ (∀𝑏 ∈
(TC‘{𝑑})𝑏 ≼ 𝑋 → ∀𝑏 ∈ (TC‘{𝑓})𝑏 ≼ 𝑋)) |
| 103 | 101, 102 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 ∈ 𝑑 → (∀𝑏 ∈ (TC‘{𝑑})𝑏 ≼ 𝑋 → ∀𝑏 ∈ (TC‘{𝑓})𝑏 ≼ 𝑋)) |
| 104 | 48, 103 | mpan9 506 |
. . . . . . . . . . . . . . 15
⊢ ((𝑑 ∈ 𝑆 ∧ 𝑓 ∈ 𝑑) → ∀𝑏 ∈ (TC‘{𝑓})𝑏 ≼ 𝑋) |
| 105 | | sneq 4636 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 = 𝑓 → {𝑎} = {𝑓}) |
| 106 | 105 | fveq2d 6910 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑓 → (TC‘{𝑎}) = (TC‘{𝑓})) |
| 107 | 106 | raleqdv 3326 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑓 → (∀𝑏 ∈ (TC‘{𝑎})𝑏 ≼ 𝑋 ↔ ∀𝑏 ∈ (TC‘{𝑓})𝑏 ≼ 𝑋)) |
| 108 | 107, 46 | elrab2 3695 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 ∈ 𝑆 ↔ (𝑓 ∈ ∪
(𝑅1 “ On) ∧ ∀𝑏 ∈ (TC‘{𝑓})𝑏 ≼ 𝑋)) |
| 109 | 95, 104, 108 | sylanbrc 583 |
. . . . . . . . . . . . . 14
⊢ ((𝑑 ∈ 𝑆 ∧ 𝑓 ∈ 𝑑) → 𝑓 ∈ 𝑆) |
| 110 | 109 | adantll 714 |
. . . . . . . . . . . . 13
⊢ (((𝑒 ∈ ω ∧ 𝑑 ∈ 𝑆) ∧ 𝑓 ∈ 𝑑) → 𝑓 ∈ 𝑆) |
| 111 | 110 | adantll 714 |
. . . . . . . . . . . 12
⊢
(((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ 𝑆)) ∧ 𝑓 ∈ 𝑑) → 𝑓 ∈ 𝑆) |
| 112 | 89, 90, 111 | rspcdva 3623 |
. . . . . . . . . . 11
⊢
(((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ 𝑆)) ∧ 𝑓 ∈ 𝑑) → dom OrdIso( E , (rank “
((𝑈‘𝑓)‘𝑒))) ∈ (𝐻‘𝑒)) |
| 113 | | imassrn 6089 |
. . . . . . . . . . . . 13
⊢ (rank
“ ((𝑈‘𝑓)‘𝑒)) ⊆ ran rank |
| 114 | 113, 32 | sstri 3993 |
. . . . . . . . . . . 12
⊢ (rank
“ ((𝑈‘𝑓)‘𝑒)) ⊆ On |
| 115 | | fvex 6919 |
. . . . . . . . . . . . . . 15
⊢ ((𝑈‘𝑓)‘𝑒) ∈ V |
| 116 | 115 | funimaex 6655 |
. . . . . . . . . . . . . 14
⊢ (Fun rank
→ (rank “ ((𝑈‘𝑓)‘𝑒)) ∈ V) |
| 117 | 39, 116 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ (rank
“ ((𝑈‘𝑓)‘𝑒)) ∈ V |
| 118 | 117 | elpw 4604 |
. . . . . . . . . . . 12
⊢ ((rank
“ ((𝑈‘𝑓)‘𝑒)) ∈ 𝒫 On ↔ (rank “
((𝑈‘𝑓)‘𝑒)) ⊆ On) |
| 119 | 114, 118 | mpbir 231 |
. . . . . . . . . . 11
⊢ (rank
“ ((𝑈‘𝑓)‘𝑒)) ∈ 𝒫 On |
| 120 | 112, 119 | jctil 519 |
. . . . . . . . . 10
⊢
(((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ 𝑆)) ∧ 𝑓 ∈ 𝑑) → ((rank “ ((𝑈‘𝑓)‘𝑒)) ∈ 𝒫 On ∧ dom OrdIso( E ,
(rank “ ((𝑈‘𝑓)‘𝑒))) ∈ (𝐻‘𝑒))) |
| 121 | 120 | ralrimiva 3146 |
. . . . . . . . 9
⊢
((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ 𝑆)) → ∀𝑓 ∈ 𝑑 ((rank “ ((𝑈‘𝑓)‘𝑒)) ∈ 𝒫 On ∧ dom OrdIso( E ,
(rank “ ((𝑈‘𝑓)‘𝑒))) ∈ (𝐻‘𝑒))) |
| 122 | | eqid 2737 |
. . . . . . . . . 10
⊢ OrdIso( E
, (rank “ ((𝑈‘𝑓)‘𝑒))) = OrdIso( E , (rank “ ((𝑈‘𝑓)‘𝑒))) |
| 123 | | eqid 2737 |
. . . . . . . . . 10
⊢ OrdIso( E
, ∪ 𝑓 ∈ 𝑑 (rank “ ((𝑈‘𝑓)‘𝑒))) = OrdIso( E , ∪ 𝑓 ∈ 𝑑 (rank “ ((𝑈‘𝑓)‘𝑒))) |
| 124 | 122, 123 | hsmexlem3 10468 |
. . . . . . . . 9
⊢ (((𝑑 ≼* 𝑋 ∧ (𝐻‘𝑒) ∈ On) ∧ ∀𝑓 ∈ 𝑑 ((rank “ ((𝑈‘𝑓)‘𝑒)) ∈ 𝒫 On ∧ dom OrdIso( E ,
(rank “ ((𝑈‘𝑓)‘𝑒))) ∈ (𝐻‘𝑒))) → dom OrdIso( E , ∪ 𝑓 ∈ 𝑑 (rank “ ((𝑈‘𝑓)‘𝑒))) ∈ (har‘𝒫 (𝑋 × (𝐻‘𝑒)))) |
| 125 | 80, 82, 121, 124 | syl21anc 838 |
. . . . . . . 8
⊢
((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ 𝑆)) → dom OrdIso( E , ∪ 𝑓 ∈ 𝑑 (rank “ ((𝑈‘𝑓)‘𝑒))) ∈ (har‘𝒫 (𝑋 × (𝐻‘𝑒)))) |
| 126 | 79, 125 | eqeltrid 2845 |
. . . . . . 7
⊢
((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ 𝑆)) → dom OrdIso( E , (rank “
((𝑈‘𝑑)‘suc 𝑒))) ∈ (har‘𝒫 (𝑋 × (𝐻‘𝑒)))) |
| 127 | 65 | hsmexlem8 10464 |
. . . . . . . 8
⊢ (𝑒 ∈ ω → (𝐻‘suc 𝑒) = (har‘𝒫 (𝑋 × (𝐻‘𝑒)))) |
| 128 | 127 | ad2antrl 728 |
. . . . . . 7
⊢
((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ 𝑆)) → (𝐻‘suc 𝑒) = (har‘𝒫 (𝑋 × (𝐻‘𝑒)))) |
| 129 | 126, 128 | eleqtrrd 2844 |
. . . . . 6
⊢
((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ 𝑆)) → dom OrdIso( E , (rank “
((𝑈‘𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)) |
| 130 | 129 | expr 456 |
. . . . 5
⊢
((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ 𝑒 ∈ ω) → (𝑑 ∈ 𝑆 → dom OrdIso( E , (rank “
((𝑈‘𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒))) |
| 131 | 71, 130 | ralrimi 3257 |
. . . 4
⊢
((∀𝑑 ∈
𝑆 dom OrdIso( E , (rank
“ ((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) ∧ 𝑒 ∈ ω) → ∀𝑑 ∈ 𝑆 dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)) |
| 132 | 131 | expcom 413 |
. . 3
⊢ (𝑒 ∈ ω →
(∀𝑑 ∈ 𝑆 dom OrdIso( E , (rank “
((𝑈‘𝑑)‘𝑒))) ∈ (𝐻‘𝑒) → ∀𝑑 ∈ 𝑆 dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒))) |
| 133 | 10, 19, 28, 68, 132 | finds1 7921 |
. 2
⊢ (𝑐 ∈ ω →
∀𝑑 ∈ 𝑆 dom 𝑂 ∈ (𝐻‘𝑐)) |
| 134 | 133 | r19.21bi 3251 |
1
⊢ ((𝑐 ∈ ω ∧ 𝑑 ∈ 𝑆) → dom 𝑂 ∈ (𝐻‘𝑐)) |