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Theorem hsmexlem4 10406
Description: Lemma for hsmex 10409. The core induction, establishing bounds on the order types of iterated unions of the initial set. (Contributed by Stefan O'Rear, 14-Feb-2015.)
Hypotheses
Ref Expression
hsmexlem4.x 𝑋 ∈ V
hsmexlem4.h 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
hsmexlem4.u 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
hsmexlem4.s 𝑆 = {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋}
hsmexlem4.o 𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))
Assertion
Ref Expression
hsmexlem4 ((𝑐 ∈ ω ∧ 𝑑𝑆) → dom 𝑂 ∈ (𝐻𝑐))
Distinct variable groups:   𝑎,𝑐,𝑑,𝐻   𝑆,𝑐,𝑑   𝑈,𝑐,𝑑   𝑎,𝑏,𝑧,𝑋   𝑥,𝑎,𝑦   𝑏,𝑐,𝑑,𝑥,𝑦,𝑧
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑧,𝑎,𝑏)   𝑈(𝑥,𝑦,𝑧,𝑎,𝑏)   𝐻(𝑥,𝑦,𝑧,𝑏)   𝑂(𝑥,𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝑋(𝑥,𝑦,𝑐,𝑑)

Proof of Theorem hsmexlem4
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hsmexlem4.o . . . . . . 7 𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))
2 fveq2 6878 . . . . . . . . 9 (𝑐 = ∅ → ((𝑈𝑑)‘𝑐) = ((𝑈𝑑)‘∅))
32imaeq2d 6049 . . . . . . . 8 (𝑐 = ∅ → (rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘∅)))
4 oieq2 9490 . . . . . . . 8 ((rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘∅)) → OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈𝑑)‘∅))))
53, 4syl 17 . . . . . . 7 (𝑐 = ∅ → OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈𝑑)‘∅))))
61, 5eqtrid 2783 . . . . . 6 (𝑐 = ∅ → 𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘∅))))
76dmeqd 5897 . . . . 5 (𝑐 = ∅ → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))))
8 fveq2 6878 . . . . 5 (𝑐 = ∅ → (𝐻𝑐) = (𝐻‘∅))
97, 8eleq12d 2826 . . . 4 (𝑐 = ∅ → (dom 𝑂 ∈ (𝐻𝑐) ↔ dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅)))
109ralbidv 3176 . . 3 (𝑐 = ∅ → (∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐) ↔ ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅)))
11 fveq2 6878 . . . . . . . . 9 (𝑐 = 𝑒 → ((𝑈𝑑)‘𝑐) = ((𝑈𝑑)‘𝑒))
1211imaeq2d 6049 . . . . . . . 8 (𝑐 = 𝑒 → (rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘𝑒)))
13 oieq2 9490 . . . . . . . 8 ((rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘𝑒)) → OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))))
1412, 13syl 17 . . . . . . 7 (𝑐 = 𝑒 → OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))))
151, 14eqtrid 2783 . . . . . 6 (𝑐 = 𝑒𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))))
1615dmeqd 5897 . . . . 5 (𝑐 = 𝑒 → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))))
17 fveq2 6878 . . . . 5 (𝑐 = 𝑒 → (𝐻𝑐) = (𝐻𝑒))
1816, 17eleq12d 2826 . . . 4 (𝑐 = 𝑒 → (dom 𝑂 ∈ (𝐻𝑐) ↔ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒)))
1918ralbidv 3176 . . 3 (𝑐 = 𝑒 → (∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐) ↔ ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒)))
20 fveq2 6878 . . . . . . . . 9 (𝑐 = suc 𝑒 → ((𝑈𝑑)‘𝑐) = ((𝑈𝑑)‘suc 𝑒))
2120imaeq2d 6049 . . . . . . . 8 (𝑐 = suc 𝑒 → (rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘suc 𝑒)))
22 oieq2 9490 . . . . . . . 8 ((rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘suc 𝑒)) → OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))))
2321, 22syl 17 . . . . . . 7 (𝑐 = suc 𝑒 → OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))))
241, 23eqtrid 2783 . . . . . 6 (𝑐 = suc 𝑒𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))))
2524dmeqd 5897 . . . . 5 (𝑐 = suc 𝑒 → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))))
26 fveq2 6878 . . . . 5 (𝑐 = suc 𝑒 → (𝐻𝑐) = (𝐻‘suc 𝑒))
2725, 26eleq12d 2826 . . . 4 (𝑐 = suc 𝑒 → (dom 𝑂 ∈ (𝐻𝑐) ↔ dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
2827ralbidv 3176 . . 3 (𝑐 = suc 𝑒 → (∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐) ↔ ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
29 imassrn 6060 . . . . . . 7 (rank “ ((𝑈𝑑)‘∅)) ⊆ ran rank
30 rankf 9771 . . . . . . . 8 rank: (𝑅1 “ On)⟶On
31 frn 6711 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → ran rank ⊆ On)
3230, 31ax-mp 5 . . . . . . 7 ran rank ⊆ On
3329, 32sstri 3987 . . . . . 6 (rank “ ((𝑈𝑑)‘∅)) ⊆ On
34 hsmexlem4.u . . . . . . . . . 10 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
3534ituni0 10395 . . . . . . . . 9 (𝑑 ∈ V → ((𝑈𝑑)‘∅) = 𝑑)
3635elv 3479 . . . . . . . 8 ((𝑈𝑑)‘∅) = 𝑑
3736imaeq2i 6047 . . . . . . 7 (rank “ ((𝑈𝑑)‘∅)) = (rank “ 𝑑)
38 ffun 6707 . . . . . . . . . 10 (rank: (𝑅1 “ On)⟶On → Fun rank)
3930, 38ax-mp 5 . . . . . . . . 9 Fun rank
40 vex 3477 . . . . . . . . 9 𝑑 ∈ V
41 wdomimag 9564 . . . . . . . . 9 ((Fun rank ∧ 𝑑 ∈ V) → (rank “ 𝑑) ≼* 𝑑)
4239, 40, 41mp2an 690 . . . . . . . 8 (rank “ 𝑑) ≼* 𝑑
43 sneq 4632 . . . . . . . . . . . . 13 (𝑎 = 𝑑 → {𝑎} = {𝑑})
4443fveq2d 6882 . . . . . . . . . . . 12 (𝑎 = 𝑑 → (TC‘{𝑎}) = (TC‘{𝑑}))
4544raleqdv 3324 . . . . . . . . . . 11 (𝑎 = 𝑑 → (∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋 ↔ ∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋))
46 hsmexlem4.s . . . . . . . . . . 11 𝑆 = {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋}
4745, 46elrab2 3682 . . . . . . . . . 10 (𝑑𝑆 ↔ (𝑑 (𝑅1 “ On) ∧ ∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋))
4847simprbi 497 . . . . . . . . 9 (𝑑𝑆 → ∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋)
49 vsnex 5422 . . . . . . . . . . . 12 {𝑑} ∈ V
50 tcid 9716 . . . . . . . . . . . 12 ({𝑑} ∈ V → {𝑑} ⊆ (TC‘{𝑑}))
5149, 50ax-mp 5 . . . . . . . . . . 11 {𝑑} ⊆ (TC‘{𝑑})
52 vsnid 4659 . . . . . . . . . . 11 𝑑 ∈ {𝑑}
5351, 52sselii 3975 . . . . . . . . . 10 𝑑 ∈ (TC‘{𝑑})
54 breq1 5144 . . . . . . . . . . 11 (𝑏 = 𝑑 → (𝑏𝑋𝑑𝑋))
5554rspcv 3605 . . . . . . . . . 10 (𝑑 ∈ (TC‘{𝑑}) → (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋𝑑𝑋))
5653, 55ax-mp 5 . . . . . . . . 9 (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋𝑑𝑋)
57 domwdom 9551 . . . . . . . . 9 (𝑑𝑋𝑑* 𝑋)
5848, 56, 573syl 18 . . . . . . . 8 (𝑑𝑆𝑑* 𝑋)
59 wdomtr 9552 . . . . . . . 8 (((rank “ 𝑑) ≼* 𝑑𝑑* 𝑋) → (rank “ 𝑑) ≼* 𝑋)
6042, 58, 59sylancr 587 . . . . . . 7 (𝑑𝑆 → (rank “ 𝑑) ≼* 𝑋)
6137, 60eqbrtrid 5176 . . . . . 6 (𝑑𝑆 → (rank “ ((𝑈𝑑)‘∅)) ≼* 𝑋)
62 eqid 2731 . . . . . . 7 OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) = OrdIso( E , (rank “ ((𝑈𝑑)‘∅)))
6362hsmexlem1 10403 . . . . . 6 (((rank “ ((𝑈𝑑)‘∅)) ⊆ On ∧ (rank “ ((𝑈𝑑)‘∅)) ≼* 𝑋) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (har‘𝒫 𝑋))
6433, 61, 63sylancr 587 . . . . 5 (𝑑𝑆 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (har‘𝒫 𝑋))
65 hsmexlem4.h . . . . . 6 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
6665hsmexlem7 10400 . . . . 5 (𝐻‘∅) = (har‘𝒫 𝑋)
6764, 66eleqtrrdi 2843 . . . 4 (𝑑𝑆 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅))
6867rgen 3062 . . 3 𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅)
69 nfra1 3280 . . . . . 6 𝑑𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒)
70 nfv 1917 . . . . . 6 𝑑 𝑒 ∈ ω
7169, 70nfan 1902 . . . . 5 𝑑(∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ 𝑒 ∈ ω)
7234ituniiun 10399 . . . . . . . . . . . . 13 (𝑑 ∈ V → ((𝑈𝑑)‘suc 𝑒) = 𝑓𝑑 ((𝑈𝑓)‘𝑒))
7372elv 3479 . . . . . . . . . . . 12 ((𝑈𝑑)‘suc 𝑒) = 𝑓𝑑 ((𝑈𝑓)‘𝑒)
7473imaeq2i 6047 . . . . . . . . . . 11 (rank “ ((𝑈𝑑)‘suc 𝑒)) = (rank “ 𝑓𝑑 ((𝑈𝑓)‘𝑒))
75 imaiun 7228 . . . . . . . . . . 11 (rank “ 𝑓𝑑 ((𝑈𝑓)‘𝑒)) = 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))
7674, 75eqtri 2759 . . . . . . . . . 10 (rank “ ((𝑈𝑑)‘suc 𝑒)) = 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))
77 oieq2 9490 . . . . . . . . . 10 ((rank “ ((𝑈𝑑)‘suc 𝑒)) = 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒)) → OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) = OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))))
7876, 77ax-mp 5 . . . . . . . . 9 OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) = OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒)))
7978dmeqi 5896 . . . . . . . 8 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) = dom OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒)))
8058ad2antll 727 . . . . . . . . 9 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → 𝑑* 𝑋)
8165hsmexlem9 10402 . . . . . . . . . 10 (𝑒 ∈ ω → (𝐻𝑒) ∈ On)
8281ad2antrl 726 . . . . . . . . 9 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → (𝐻𝑒) ∈ On)
83 fveq2 6878 . . . . . . . . . . . . . . . . 17 (𝑑 = 𝑓 → (𝑈𝑑) = (𝑈𝑓))
8483fveq1d 6880 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑓 → ((𝑈𝑑)‘𝑒) = ((𝑈𝑓)‘𝑒))
8584imaeq2d 6049 . . . . . . . . . . . . . . 15 (𝑑 = 𝑓 → (rank “ ((𝑈𝑑)‘𝑒)) = (rank “ ((𝑈𝑓)‘𝑒)))
86 oieq2 9490 . . . . . . . . . . . . . . 15 ((rank “ ((𝑈𝑑)‘𝑒)) = (rank “ ((𝑈𝑓)‘𝑒)) → OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) = OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))))
8785, 86syl 17 . . . . . . . . . . . . . 14 (𝑑 = 𝑓 → OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) = OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))))
8887dmeqd 5897 . . . . . . . . . . . . 13 (𝑑 = 𝑓 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) = dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))))
8988eleq1d 2817 . . . . . . . . . . . 12 (𝑑 = 𝑓 → (dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ↔ dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) ∈ (𝐻𝑒)))
90 simpll 765 . . . . . . . . . . . 12 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒))
9146ssrab3 4076 . . . . . . . . . . . . . . . . . 18 𝑆 (𝑅1 “ On)
9291sseli 3974 . . . . . . . . . . . . . . . . 17 (𝑑𝑆𝑑 (𝑅1 “ On))
93 r1elssi 9782 . . . . . . . . . . . . . . . . 17 (𝑑 (𝑅1 “ On) → 𝑑 (𝑅1 “ On))
9492, 93syl 17 . . . . . . . . . . . . . . . 16 (𝑑𝑆𝑑 (𝑅1 “ On))
9594sselda 3978 . . . . . . . . . . . . . . 15 ((𝑑𝑆𝑓𝑑) → 𝑓 (𝑅1 “ On))
96 snssi 4804 . . . . . . . . . . . . . . . . . . 19 (𝑓𝑑 → {𝑓} ⊆ 𝑑)
9740tcss 9721 . . . . . . . . . . . . . . . . . . 19 ({𝑓} ⊆ 𝑑 → (TC‘{𝑓}) ⊆ (TC‘𝑑))
9896, 97syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓𝑑 → (TC‘{𝑓}) ⊆ (TC‘𝑑))
9949tcel 9722 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∈ {𝑑} → (TC‘𝑑) ⊆ (TC‘{𝑑}))
10052, 99mp1i 13 . . . . . . . . . . . . . . . . . 18 (𝑓𝑑 → (TC‘𝑑) ⊆ (TC‘{𝑑}))
10198, 100sstrd 3988 . . . . . . . . . . . . . . . . 17 (𝑓𝑑 → (TC‘{𝑓}) ⊆ (TC‘{𝑑}))
102 ssralv 4046 . . . . . . . . . . . . . . . . 17 ((TC‘{𝑓}) ⊆ (TC‘{𝑑}) → (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋 → ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
103101, 102syl 17 . . . . . . . . . . . . . . . 16 (𝑓𝑑 → (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋 → ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
10448, 103mpan9 507 . . . . . . . . . . . . . . 15 ((𝑑𝑆𝑓𝑑) → ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋)
105 sneq 4632 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑓 → {𝑎} = {𝑓})
106105fveq2d 6882 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑓 → (TC‘{𝑎}) = (TC‘{𝑓}))
107106raleqdv 3324 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑓 → (∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋 ↔ ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
108107, 46elrab2 3682 . . . . . . . . . . . . . . 15 (𝑓𝑆 ↔ (𝑓 (𝑅1 “ On) ∧ ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
10995, 104, 108sylanbrc 583 . . . . . . . . . . . . . 14 ((𝑑𝑆𝑓𝑑) → 𝑓𝑆)
110109adantll 712 . . . . . . . . . . . . 13 (((𝑒 ∈ ω ∧ 𝑑𝑆) ∧ 𝑓𝑑) → 𝑓𝑆)
111110adantll 712 . . . . . . . . . . . 12 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → 𝑓𝑆)
11289, 90, 111rspcdva 3610 . . . . . . . . . . 11 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) ∈ (𝐻𝑒))
113 imassrn 6060 . . . . . . . . . . . . 13 (rank “ ((𝑈𝑓)‘𝑒)) ⊆ ran rank
114113, 32sstri 3987 . . . . . . . . . . . 12 (rank “ ((𝑈𝑓)‘𝑒)) ⊆ On
115 fvex 6891 . . . . . . . . . . . . . . 15 ((𝑈𝑓)‘𝑒) ∈ V
116115funimaex 6625 . . . . . . . . . . . . . 14 (Fun rank → (rank “ ((𝑈𝑓)‘𝑒)) ∈ V)
11739, 116ax-mp 5 . . . . . . . . . . . . 13 (rank “ ((𝑈𝑓)‘𝑒)) ∈ V
118117elpw 4600 . . . . . . . . . . . 12 ((rank “ ((𝑈𝑓)‘𝑒)) ∈ 𝒫 On ↔ (rank “ ((𝑈𝑓)‘𝑒)) ⊆ On)
119114, 118mpbir 230 . . . . . . . . . . 11 (rank “ ((𝑈𝑓)‘𝑒)) ∈ 𝒫 On
120112, 119jctil 520 . . . . . . . . . 10 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → ((rank “ ((𝑈𝑓)‘𝑒)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) ∈ (𝐻𝑒)))
121120ralrimiva 3145 . . . . . . . . 9 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → ∀𝑓𝑑 ((rank “ ((𝑈𝑓)‘𝑒)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) ∈ (𝐻𝑒)))
122 eqid 2731 . . . . . . . . . 10 OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) = OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒)))
123 eqid 2731 . . . . . . . . . 10 OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))) = OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒)))
124122, 123hsmexlem3 10405 . . . . . . . . 9 (((𝑑* 𝑋 ∧ (𝐻𝑒) ∈ On) ∧ ∀𝑓𝑑 ((rank “ ((𝑈𝑓)‘𝑒)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) ∈ (𝐻𝑒))) → dom OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))) ∈ (har‘𝒫 (𝑋 × (𝐻𝑒))))
12580, 82, 121, 124syl21anc 836 . . . . . . . 8 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → dom OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))) ∈ (har‘𝒫 (𝑋 × (𝐻𝑒))))
12679, 125eqeltrid 2836 . . . . . . 7 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (har‘𝒫 (𝑋 × (𝐻𝑒))))
12765hsmexlem8 10401 . . . . . . . 8 (𝑒 ∈ ω → (𝐻‘suc 𝑒) = (har‘𝒫 (𝑋 × (𝐻𝑒))))
128127ad2antrl 726 . . . . . . 7 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → (𝐻‘suc 𝑒) = (har‘𝒫 (𝑋 × (𝐻𝑒))))
129126, 128eleqtrrd 2835 . . . . . 6 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒))
130129expr 457 . . . . 5 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ 𝑒 ∈ ω) → (𝑑𝑆 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
13171, 130ralrimi 3253 . . . 4 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ 𝑒 ∈ ω) → ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒))
132131expcom 414 . . 3 (𝑒 ∈ ω → (∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) → ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
13310, 19, 28, 68, 132finds1 7874 . 2 (𝑐 ∈ ω → ∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐))
134133r19.21bi 3247 1 ((𝑐 ∈ ω ∧ 𝑑𝑆) → dom 𝑂 ∈ (𝐻𝑐))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3060  {crab 3431  Vcvv 3473  wss 3944  c0 4318  𝒫 cpw 4596  {csn 4622   cuni 4901   ciun 4990   class class class wbr 5141  cmpt 5224   E cep 5572   × cxp 5667  dom cdm 5669  ran crn 5670  cres 5671  cima 5672  Oncon0 6353  suc csuc 6355  Fun wfun 6526  wf 6528  cfv 6532  ωcom 7838  reccrdg 8391  cdom 8920  OrdIsocoi 9486  harchar 9533  * cwdom 9541  TCctc 9713  𝑅1cr1 9739  rankcrnk 9740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-inf2 9618
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-smo 8328  df-recs 8353  df-rdg 8392  df-en 8923  df-dom 8924  df-sdom 8925  df-oi 9487  df-har 9534  df-wdom 9542  df-tc 9714  df-r1 9741  df-rank 9742
This theorem is referenced by:  hsmexlem5  10407
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