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Theorem hsmexlem4 10469
Description: Lemma for hsmex 10472. 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 6906 . . . . . . . . 9 (𝑐 = ∅ → ((𝑈𝑑)‘𝑐) = ((𝑈𝑑)‘∅))
32imaeq2d 6078 . . . . . . . 8 (𝑐 = ∅ → (rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘∅)))
4 oieq2 9553 . . . . . . . 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 2789 . . . . . 6 (𝑐 = ∅ → 𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘∅))))
76dmeqd 5916 . . . . 5 (𝑐 = ∅ → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))))
8 fveq2 6906 . . . . 5 (𝑐 = ∅ → (𝐻𝑐) = (𝐻‘∅))
97, 8eleq12d 2835 . . . 4 (𝑐 = ∅ → (dom 𝑂 ∈ (𝐻𝑐) ↔ dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅)))
109ralbidv 3178 . . 3 (𝑐 = ∅ → (∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐) ↔ ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅)))
11 fveq2 6906 . . . . . . . . 9 (𝑐 = 𝑒 → ((𝑈𝑑)‘𝑐) = ((𝑈𝑑)‘𝑒))
1211imaeq2d 6078 . . . . . . . 8 (𝑐 = 𝑒 → (rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘𝑒)))
13 oieq2 9553 . . . . . . . 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 2789 . . . . . 6 (𝑐 = 𝑒𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))))
1615dmeqd 5916 . . . . 5 (𝑐 = 𝑒 → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))))
17 fveq2 6906 . . . . 5 (𝑐 = 𝑒 → (𝐻𝑐) = (𝐻𝑒))
1816, 17eleq12d 2835 . . . 4 (𝑐 = 𝑒 → (dom 𝑂 ∈ (𝐻𝑐) ↔ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒)))
1918ralbidv 3178 . . 3 (𝑐 = 𝑒 → (∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐) ↔ ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒)))
20 fveq2 6906 . . . . . . . . 9 (𝑐 = suc 𝑒 → ((𝑈𝑑)‘𝑐) = ((𝑈𝑑)‘suc 𝑒))
2120imaeq2d 6078 . . . . . . . 8 (𝑐 = suc 𝑒 → (rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘suc 𝑒)))
22 oieq2 9553 . . . . . . . 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 2789 . . . . . 6 (𝑐 = suc 𝑒𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))))
2524dmeqd 5916 . . . . 5 (𝑐 = suc 𝑒 → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))))
26 fveq2 6906 . . . . 5 (𝑐 = suc 𝑒 → (𝐻𝑐) = (𝐻‘suc 𝑒))
2725, 26eleq12d 2835 . . . 4 (𝑐 = suc 𝑒 → (dom 𝑂 ∈ (𝐻𝑐) ↔ dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
2827ralbidv 3178 . . 3 (𝑐 = suc 𝑒 → (∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐) ↔ ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
29 imassrn 6089 . . . . . . 7 (rank “ ((𝑈𝑑)‘∅)) ⊆ ran rank
30 rankf 9834 . . . . . . . 8 rank: (𝑅1 “ On)⟶On
31 frn 6743 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → ran rank ⊆ On)
3230, 31ax-mp 5 . . . . . . 7 ran rank ⊆ On
3329, 32sstri 3993 . . . . . 6 (rank “ ((𝑈𝑑)‘∅)) ⊆ On
34 hsmexlem4.u . . . . . . . . . 10 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
3534ituni0 10458 . . . . . . . . 9 (𝑑 ∈ V → ((𝑈𝑑)‘∅) = 𝑑)
3635elv 3485 . . . . . . . 8 ((𝑈𝑑)‘∅) = 𝑑
3736imaeq2i 6076 . . . . . . 7 (rank “ ((𝑈𝑑)‘∅)) = (rank “ 𝑑)
38 ffun 6739 . . . . . . . . . 10 (rank: (𝑅1 “ On)⟶On → Fun rank)
3930, 38ax-mp 5 . . . . . . . . 9 Fun rank
40 vex 3484 . . . . . . . . 9 𝑑 ∈ V
41 wdomimag 9627 . . . . . . . . 9 ((Fun rank ∧ 𝑑 ∈ V) → (rank “ 𝑑) ≼* 𝑑)
4239, 40, 41mp2an 692 . . . . . . . 8 (rank “ 𝑑) ≼* 𝑑
43 sneq 4636 . . . . . . . . . . . . 13 (𝑎 = 𝑑 → {𝑎} = {𝑑})
4443fveq2d 6910 . . . . . . . . . . . 12 (𝑎 = 𝑑 → (TC‘{𝑎}) = (TC‘{𝑑}))
4544raleqdv 3326 . . . . . . . . . . 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 3695 . . . . . . . . . 10 (𝑑𝑆 ↔ (𝑑 (𝑅1 “ On) ∧ ∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋))
4847simprbi 496 . . . . . . . . 9 (𝑑𝑆 → ∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋)
49 vsnex 5434 . . . . . . . . . . . 12 {𝑑} ∈ V
50 tcid 9779 . . . . . . . . . . . 12 ({𝑑} ∈ V → {𝑑} ⊆ (TC‘{𝑑}))
5149, 50ax-mp 5 . . . . . . . . . . 11 {𝑑} ⊆ (TC‘{𝑑})
52 vsnid 4663 . . . . . . . . . . 11 𝑑 ∈ {𝑑}
5351, 52sselii 3980 . . . . . . . . . 10 𝑑 ∈ (TC‘{𝑑})
54 breq1 5146 . . . . . . . . . . 11 (𝑏 = 𝑑 → (𝑏𝑋𝑑𝑋))
5554rspcv 3618 . . . . . . . . . 10 (𝑑 ∈ (TC‘{𝑑}) → (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋𝑑𝑋))
5653, 55ax-mp 5 . . . . . . . . 9 (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋𝑑𝑋)
57 domwdom 9614 . . . . . . . . 9 (𝑑𝑋𝑑* 𝑋)
5848, 56, 573syl 18 . . . . . . . 8 (𝑑𝑆𝑑* 𝑋)
59 wdomtr 9615 . . . . . . . 8 (((rank “ 𝑑) ≼* 𝑑𝑑* 𝑋) → (rank “ 𝑑) ≼* 𝑋)
6042, 58, 59sylancr 587 . . . . . . 7 (𝑑𝑆 → (rank “ 𝑑) ≼* 𝑋)
6137, 60eqbrtrid 5178 . . . . . 6 (𝑑𝑆 → (rank “ ((𝑈𝑑)‘∅)) ≼* 𝑋)
62 eqid 2737 . . . . . . 7 OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) = OrdIso( E , (rank “ ((𝑈𝑑)‘∅)))
6362hsmexlem1 10466 . . . . . 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 10463 . . . . 5 (𝐻‘∅) = (har‘𝒫 𝑋)
6764, 66eleqtrrdi 2852 . . . 4 (𝑑𝑆 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅))
6867rgen 3063 . . 3 𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅)
69 nfra1 3284 . . . . . 6 𝑑𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒)
70 nfv 1914 . . . . . 6 𝑑 𝑒 ∈ ω
7169, 70nfan 1899 . . . . 5 𝑑(∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ 𝑒 ∈ ω)
7234ituniiun 10462 . . . . . . . . . . . . 13 (𝑑 ∈ V → ((𝑈𝑑)‘suc 𝑒) = 𝑓𝑑 ((𝑈𝑓)‘𝑒))
7372elv 3485 . . . . . . . . . . . 12 ((𝑈𝑑)‘suc 𝑒) = 𝑓𝑑 ((𝑈𝑓)‘𝑒)
7473imaeq2i 6076 . . . . . . . . . . 11 (rank “ ((𝑈𝑑)‘suc 𝑒)) = (rank “ 𝑓𝑑 ((𝑈𝑓)‘𝑒))
75 imaiun 7265 . . . . . . . . . . 11 (rank “ 𝑓𝑑 ((𝑈𝑓)‘𝑒)) = 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))
7674, 75eqtri 2765 . . . . . . . . . 10 (rank “ ((𝑈𝑑)‘suc 𝑒)) = 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))
77 oieq2 9553 . . . . . . . . . 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 5915 . . . . . . . 8 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) = dom OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒)))
8058ad2antll 729 . . . . . . . . 9 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → 𝑑* 𝑋)
8165hsmexlem9 10465 . . . . . . . . . 10 (𝑒 ∈ ω → (𝐻𝑒) ∈ On)
8281ad2antrl 728 . . . . . . . . 9 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → (𝐻𝑒) ∈ On)
83 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑑 = 𝑓 → (𝑈𝑑) = (𝑈𝑓))
8483fveq1d 6908 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑓 → ((𝑈𝑑)‘𝑒) = ((𝑈𝑓)‘𝑒))
8584imaeq2d 6078 . . . . . . . . . . . . . . 15 (𝑑 = 𝑓 → (rank “ ((𝑈𝑑)‘𝑒)) = (rank “ ((𝑈𝑓)‘𝑒)))
86 oieq2 9553 . . . . . . . . . . . . . . 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 5916 . . . . . . . . . . . . 13 (𝑑 = 𝑓 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) = dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))))
8988eleq1d 2826 . . . . . . . . . . . 12 (𝑑 = 𝑓 → (dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ↔ dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) ∈ (𝐻𝑒)))
90 simpll 767 . . . . . . . . . . . 12 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒))
9146ssrab3 4082 . . . . . . . . . . . . . . . . . 18 𝑆 (𝑅1 “ On)
9291sseli 3979 . . . . . . . . . . . . . . . . 17 (𝑑𝑆𝑑 (𝑅1 “ On))
93 r1elssi 9845 . . . . . . . . . . . . . . . . 17 (𝑑 (𝑅1 “ On) → 𝑑 (𝑅1 “ On))
9492, 93syl 17 . . . . . . . . . . . . . . . 16 (𝑑𝑆𝑑 (𝑅1 “ On))
9594sselda 3983 . . . . . . . . . . . . . . 15 ((𝑑𝑆𝑓𝑑) → 𝑓 (𝑅1 “ On))
96 snssi 4808 . . . . . . . . . . . . . . . . . . 19 (𝑓𝑑 → {𝑓} ⊆ 𝑑)
9740tcss 9784 . . . . . . . . . . . . . . . . . . 19 ({𝑓} ⊆ 𝑑 → (TC‘{𝑓}) ⊆ (TC‘𝑑))
9896, 97syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓𝑑 → (TC‘{𝑓}) ⊆ (TC‘𝑑))
9949tcel 9785 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∈ {𝑑} → (TC‘𝑑) ⊆ (TC‘{𝑑}))
10052, 99mp1i 13 . . . . . . . . . . . . . . . . . 18 (𝑓𝑑 → (TC‘𝑑) ⊆ (TC‘{𝑑}))
10198, 100sstrd 3994 . . . . . . . . . . . . . . . . 17 (𝑓𝑑 → (TC‘{𝑓}) ⊆ (TC‘{𝑑}))
102 ssralv 4052 . . . . . . . . . . . . . . . . 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 506 . . . . . . . . . . . . . . 15 ((𝑑𝑆𝑓𝑑) → ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋)
105 sneq 4636 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑓 → {𝑎} = {𝑓})
106105fveq2d 6910 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑓 → (TC‘{𝑎}) = (TC‘{𝑓}))
107106raleqdv 3326 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑓 → (∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋 ↔ ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
108107, 46elrab2 3695 . . . . . . . . . . . . . . 15 (𝑓𝑆 ↔ (𝑓 (𝑅1 “ On) ∧ ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
10995, 104, 108sylanbrc 583 . . . . . . . . . . . . . 14 ((𝑑𝑆𝑓𝑑) → 𝑓𝑆)
110109adantll 714 . . . . . . . . . . . . 13 (((𝑒 ∈ ω ∧ 𝑑𝑆) ∧ 𝑓𝑑) → 𝑓𝑆)
111110adantll 714 . . . . . . . . . . . 12 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → 𝑓𝑆)
11289, 90, 111rspcdva 3623 . . . . . . . . . . 11 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) ∈ (𝐻𝑒))
113 imassrn 6089 . . . . . . . . . . . . 13 (rank “ ((𝑈𝑓)‘𝑒)) ⊆ ran rank
114113, 32sstri 3993 . . . . . . . . . . . 12 (rank “ ((𝑈𝑓)‘𝑒)) ⊆ On
115 fvex 6919 . . . . . . . . . . . . . . 15 ((𝑈𝑓)‘𝑒) ∈ V
116115funimaex 6655 . . . . . . . . . . . . . 14 (Fun rank → (rank “ ((𝑈𝑓)‘𝑒)) ∈ V)
11739, 116ax-mp 5 . . . . . . . . . . . . 13 (rank “ ((𝑈𝑓)‘𝑒)) ∈ V
118117elpw 4604 . . . . . . . . . . . 12 ((rank “ ((𝑈𝑓)‘𝑒)) ∈ 𝒫 On ↔ (rank “ ((𝑈𝑓)‘𝑒)) ⊆ On)
119114, 118mpbir 231 . . . . . . . . . . 11 (rank “ ((𝑈𝑓)‘𝑒)) ∈ 𝒫 On
120112, 119jctil 519 . . . . . . . . . 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 3146 . . . . . . . . 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 2737 . . . . . . . . . 10 OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) = OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒)))
123 eqid 2737 . . . . . . . . . 10 OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))) = OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒)))
124122, 123hsmexlem3 10468 . . . . . . . . 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 838 . . . . . . . 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 2845 . . . . . . 7 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (har‘𝒫 (𝑋 × (𝐻𝑒))))
12765hsmexlem8 10464 . . . . . . . 8 (𝑒 ∈ ω → (𝐻‘suc 𝑒) = (har‘𝒫 (𝑋 × (𝐻𝑒))))
128127ad2antrl 728 . . . . . . 7 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → (𝐻‘suc 𝑒) = (har‘𝒫 (𝑋 × (𝐻𝑒))))
129126, 128eleqtrrd 2844 . . . . . 6 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒))
130129expr 456 . . . . 5 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ 𝑒 ∈ ω) → (𝑑𝑆 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
13171, 130ralrimi 3257 . . . 4 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ 𝑒 ∈ ω) → ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒))
132131expcom 413 . . 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 7921 . 2 (𝑐 ∈ ω → ∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐))
134133r19.21bi 3251 1 ((𝑐 ∈ ω ∧ 𝑑𝑆) → dom 𝑂 ∈ (𝐻𝑐))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   cuni 4907   ciun 4991   class class class wbr 5143  cmpt 5225   E cep 5583   × cxp 5683  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Oncon0 6384  suc csuc 6386  Fun wfun 6555  wf 6557  cfv 6561  ωcom 7887  reccrdg 8449  cdom 8983  OrdIsocoi 9549  harchar 9596  * cwdom 9604  TCctc 9776  𝑅1cr1 9802  rankcrnk 9803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-smo 8386  df-recs 8411  df-rdg 8450  df-en 8986  df-dom 8987  df-sdom 8988  df-oi 9550  df-har 9597  df-wdom 9605  df-tc 9777  df-r1 9804  df-rank 9805
This theorem is referenced by:  hsmexlem5  10470
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