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Theorem hsmexlem4 9134
 Description: Lemma for hsmex 9137. 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 6103 . . . . . . . . 9 (𝑐 = ∅ → ((𝑈𝑑)‘𝑐) = ((𝑈𝑑)‘∅))
32imaeq2d 5385 . . . . . . . 8 (𝑐 = ∅ → (rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘∅)))
4 oieq2 8301 . . . . . . . 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, 5syl5eq 2656 . . . . . 6 (𝑐 = ∅ → 𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘∅))))
76dmeqd 5248 . . . . 5 (𝑐 = ∅ → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))))
8 fveq2 6103 . . . . 5 (𝑐 = ∅ → (𝐻𝑐) = (𝐻‘∅))
97, 8eleq12d 2682 . . . 4 (𝑐 = ∅ → (dom 𝑂 ∈ (𝐻𝑐) ↔ dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅)))
109ralbidv 2969 . . 3 (𝑐 = ∅ → (∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐) ↔ ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅)))
11 fveq2 6103 . . . . . . . . 9 (𝑐 = 𝑒 → ((𝑈𝑑)‘𝑐) = ((𝑈𝑑)‘𝑒))
1211imaeq2d 5385 . . . . . . . 8 (𝑐 = 𝑒 → (rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘𝑒)))
13 oieq2 8301 . . . . . . . 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, 14syl5eq 2656 . . . . . 6 (𝑐 = 𝑒𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))))
1615dmeqd 5248 . . . . 5 (𝑐 = 𝑒 → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))))
17 fveq2 6103 . . . . 5 (𝑐 = 𝑒 → (𝐻𝑐) = (𝐻𝑒))
1816, 17eleq12d 2682 . . . 4 (𝑐 = 𝑒 → (dom 𝑂 ∈ (𝐻𝑐) ↔ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒)))
1918ralbidv 2969 . . 3 (𝑐 = 𝑒 → (∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐) ↔ ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒)))
20 fveq2 6103 . . . . . . . . 9 (𝑐 = suc 𝑒 → ((𝑈𝑑)‘𝑐) = ((𝑈𝑑)‘suc 𝑒))
2120imaeq2d 5385 . . . . . . . 8 (𝑐 = suc 𝑒 → (rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘suc 𝑒)))
22 oieq2 8301 . . . . . . . 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, 23syl5eq 2656 . . . . . 6 (𝑐 = suc 𝑒𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))))
2524dmeqd 5248 . . . . 5 (𝑐 = suc 𝑒 → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))))
26 fveq2 6103 . . . . 5 (𝑐 = suc 𝑒 → (𝐻𝑐) = (𝐻‘suc 𝑒))
2725, 26eleq12d 2682 . . . 4 (𝑐 = suc 𝑒 → (dom 𝑂 ∈ (𝐻𝑐) ↔ dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
2827ralbidv 2969 . . 3 (𝑐 = suc 𝑒 → (∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐) ↔ ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
29 imassrn 5396 . . . . . . 7 (rank “ ((𝑈𝑑)‘∅)) ⊆ ran rank
30 rankf 8540 . . . . . . . 8 rank: (𝑅1 “ On)⟶On
31 frn 5966 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → ran rank ⊆ On)
3230, 31ax-mp 5 . . . . . . 7 ran rank ⊆ On
3329, 32sstri 3577 . . . . . 6 (rank “ ((𝑈𝑑)‘∅)) ⊆ On
34 vex 3176 . . . . . . . . 9 𝑑 ∈ V
35 hsmexlem4.u . . . . . . . . . 10 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
3635ituni0 9123 . . . . . . . . 9 (𝑑 ∈ V → ((𝑈𝑑)‘∅) = 𝑑)
3734, 36ax-mp 5 . . . . . . . 8 ((𝑈𝑑)‘∅) = 𝑑
3837imaeq2i 5383 . . . . . . 7 (rank “ ((𝑈𝑑)‘∅)) = (rank “ 𝑑)
39 ffun 5961 . . . . . . . . . 10 (rank: (𝑅1 “ On)⟶On → Fun rank)
4030, 39ax-mp 5 . . . . . . . . 9 Fun rank
41 wdomimag 8375 . . . . . . . . 9 ((Fun rank ∧ 𝑑 ∈ V) → (rank “ 𝑑) ≼* 𝑑)
4240, 34, 41mp2an 704 . . . . . . . 8 (rank “ 𝑑) ≼* 𝑑
43 sneq 4135 . . . . . . . . . . . . 13 (𝑎 = 𝑑 → {𝑎} = {𝑑})
4443fveq2d 6107 . . . . . . . . . . . 12 (𝑎 = 𝑑 → (TC‘{𝑎}) = (TC‘{𝑑}))
4544raleqdv 3121 . . . . . . . . . . 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 3333 . . . . . . . . . 10 (𝑑𝑆 ↔ (𝑑 (𝑅1 “ On) ∧ ∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋))
4847simprbi 479 . . . . . . . . 9 (𝑑𝑆 → ∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋)
49 snex 4835 . . . . . . . . . . . 12 {𝑑} ∈ V
50 tcid 8498 . . . . . . . . . . . 12 ({𝑑} ∈ V → {𝑑} ⊆ (TC‘{𝑑}))
5149, 50ax-mp 5 . . . . . . . . . . 11 {𝑑} ⊆ (TC‘{𝑑})
52 vsnid 4156 . . . . . . . . . . 11 𝑑 ∈ {𝑑}
5351, 52sselii 3565 . . . . . . . . . 10 𝑑 ∈ (TC‘{𝑑})
54 breq1 4586 . . . . . . . . . . 11 (𝑏 = 𝑑 → (𝑏𝑋𝑑𝑋))
5554rspcv 3278 . . . . . . . . . 10 (𝑑 ∈ (TC‘{𝑑}) → (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋𝑑𝑋))
5653, 55ax-mp 5 . . . . . . . . 9 (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋𝑑𝑋)
57 domwdom 8362 . . . . . . . . 9 (𝑑𝑋𝑑* 𝑋)
5848, 56, 573syl 18 . . . . . . . 8 (𝑑𝑆𝑑* 𝑋)
59 wdomtr 8363 . . . . . . . 8 (((rank “ 𝑑) ≼* 𝑑𝑑* 𝑋) → (rank “ 𝑑) ≼* 𝑋)
6042, 58, 59sylancr 694 . . . . . . 7 (𝑑𝑆 → (rank “ 𝑑) ≼* 𝑋)
6138, 60syl5eqbr 4618 . . . . . 6 (𝑑𝑆 → (rank “ ((𝑈𝑑)‘∅)) ≼* 𝑋)
62 eqid 2610 . . . . . . 7 OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) = OrdIso( E , (rank “ ((𝑈𝑑)‘∅)))
6362hsmexlem1 9131 . . . . . 6 (((rank “ ((𝑈𝑑)‘∅)) ⊆ On ∧ (rank “ ((𝑈𝑑)‘∅)) ≼* 𝑋) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (har‘𝒫 𝑋))
6433, 61, 63sylancr 694 . . . . 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 9128 . . . . 5 (𝐻‘∅) = (har‘𝒫 𝑋)
6764, 66syl6eleqr 2699 . . . 4 (𝑑𝑆 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅))
6867rgen 2906 . . 3 𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅)
69 nfra1 2925 . . . . . 6 𝑑𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒)
70 nfv 1830 . . . . . 6 𝑑 𝑒 ∈ ω
7169, 70nfan 1816 . . . . 5 𝑑(∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ 𝑒 ∈ ω)
7235ituniiun 9127 . . . . . . . . . . . . 13 (𝑑 ∈ V → ((𝑈𝑑)‘suc 𝑒) = 𝑓𝑑 ((𝑈𝑓)‘𝑒))
7334, 72ax-mp 5 . . . . . . . . . . . 12 ((𝑈𝑑)‘suc 𝑒) = 𝑓𝑑 ((𝑈𝑓)‘𝑒)
7473imaeq2i 5383 . . . . . . . . . . 11 (rank “ ((𝑈𝑑)‘suc 𝑒)) = (rank “ 𝑓𝑑 ((𝑈𝑓)‘𝑒))
75 imaiun 6407 . . . . . . . . . . 11 (rank “ 𝑓𝑑 ((𝑈𝑓)‘𝑒)) = 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))
7674, 75eqtri 2632 . . . . . . . . . 10 (rank “ ((𝑈𝑑)‘suc 𝑒)) = 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))
77 oieq2 8301 . . . . . . . . . 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 5247 . . . . . . . 8 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) = dom OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒)))
8058ad2antll 761 . . . . . . . . 9 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → 𝑑* 𝑋)
8165hsmexlem9 9130 . . . . . . . . . 10 (𝑒 ∈ ω → (𝐻𝑒) ∈ On)
8281ad2antrl 760 . . . . . . . . 9 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → (𝐻𝑒) ∈ On)
83 ssrab2 3650 . . . . . . . . . . . . . . . . . . 19 {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋} ⊆ (𝑅1 “ On)
8446, 83eqsstri 3598 . . . . . . . . . . . . . . . . . 18 𝑆 (𝑅1 “ On)
8584sseli 3564 . . . . . . . . . . . . . . . . 17 (𝑑𝑆𝑑 (𝑅1 “ On))
86 r1elssi 8551 . . . . . . . . . . . . . . . . 17 (𝑑 (𝑅1 “ On) → 𝑑 (𝑅1 “ On))
8785, 86syl 17 . . . . . . . . . . . . . . . 16 (𝑑𝑆𝑑 (𝑅1 “ On))
8887sselda 3568 . . . . . . . . . . . . . . 15 ((𝑑𝑆𝑓𝑑) → 𝑓 (𝑅1 “ On))
89 snssi 4280 . . . . . . . . . . . . . . . . . . 19 (𝑓𝑑 → {𝑓} ⊆ 𝑑)
9034tcss 8503 . . . . . . . . . . . . . . . . . . 19 ({𝑓} ⊆ 𝑑 → (TC‘{𝑓}) ⊆ (TC‘𝑑))
9189, 90syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓𝑑 → (TC‘{𝑓}) ⊆ (TC‘𝑑))
9249tcel 8504 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∈ {𝑑} → (TC‘𝑑) ⊆ (TC‘{𝑑}))
9352, 92mp1i 13 . . . . . . . . . . . . . . . . . 18 (𝑓𝑑 → (TC‘𝑑) ⊆ (TC‘{𝑑}))
9491, 93sstrd 3578 . . . . . . . . . . . . . . . . 17 (𝑓𝑑 → (TC‘{𝑓}) ⊆ (TC‘{𝑑}))
95 ssralv 3629 . . . . . . . . . . . . . . . . 17 ((TC‘{𝑓}) ⊆ (TC‘{𝑑}) → (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋 → ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
9694, 95syl 17 . . . . . . . . . . . . . . . 16 (𝑓𝑑 → (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋 → ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
9748, 96mpan9 485 . . . . . . . . . . . . . . 15 ((𝑑𝑆𝑓𝑑) → ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋)
98 sneq 4135 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑓 → {𝑎} = {𝑓})
9998fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑓 → (TC‘{𝑎}) = (TC‘{𝑓}))
10099raleqdv 3121 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑓 → (∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋 ↔ ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
101100, 46elrab2 3333 . . . . . . . . . . . . . . 15 (𝑓𝑆 ↔ (𝑓 (𝑅1 “ On) ∧ ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
10288, 97, 101sylanbrc 695 . . . . . . . . . . . . . 14 ((𝑑𝑆𝑓𝑑) → 𝑓𝑆)
103102adantll 746 . . . . . . . . . . . . 13 (((𝑒 ∈ ω ∧ 𝑑𝑆) ∧ 𝑓𝑑) → 𝑓𝑆)
104103adantll 746 . . . . . . . . . . . 12 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → 𝑓𝑆)
105 simpll 786 . . . . . . . . . . . 12 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒))
106 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑑 = 𝑓 → (𝑈𝑑) = (𝑈𝑓))
107106fveq1d 6105 . . . . . . . . . . . . . . . . 17 (𝑑 = 𝑓 → ((𝑈𝑑)‘𝑒) = ((𝑈𝑓)‘𝑒))
108107imaeq2d 5385 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑓 → (rank “ ((𝑈𝑑)‘𝑒)) = (rank “ ((𝑈𝑓)‘𝑒)))
109 oieq2 8301 . . . . . . . . . . . . . . . 16 ((rank “ ((𝑈𝑑)‘𝑒)) = (rank “ ((𝑈𝑓)‘𝑒)) → OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) = OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))))
110108, 109syl 17 . . . . . . . . . . . . . . 15 (𝑑 = 𝑓 → OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) = OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))))
111110dmeqd 5248 . . . . . . . . . . . . . 14 (𝑑 = 𝑓 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) = dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))))
112111eleq1d 2672 . . . . . . . . . . . . 13 (𝑑 = 𝑓 → (dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ↔ dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) ∈ (𝐻𝑒)))
113112rspcv 3278 . . . . . . . . . . . 12 (𝑓𝑆 → (∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) → dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) ∈ (𝐻𝑒)))
114104, 105, 113sylc 63 . . . . . . . . . . 11 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) ∈ (𝐻𝑒))
115 imassrn 5396 . . . . . . . . . . . . 13 (rank “ ((𝑈𝑓)‘𝑒)) ⊆ ran rank
116115, 32sstri 3577 . . . . . . . . . . . 12 (rank “ ((𝑈𝑓)‘𝑒)) ⊆ On
117 fvex 6113 . . . . . . . . . . . . . . 15 ((𝑈𝑓)‘𝑒) ∈ V
118117funimaex 5890 . . . . . . . . . . . . . 14 (Fun rank → (rank “ ((𝑈𝑓)‘𝑒)) ∈ V)
11940, 118ax-mp 5 . . . . . . . . . . . . 13 (rank “ ((𝑈𝑓)‘𝑒)) ∈ V
120119elpw 4114 . . . . . . . . . . . 12 ((rank “ ((𝑈𝑓)‘𝑒)) ∈ 𝒫 On ↔ (rank “ ((𝑈𝑓)‘𝑒)) ⊆ On)
121116, 120mpbir 220 . . . . . . . . . . 11 (rank “ ((𝑈𝑓)‘𝑒)) ∈ 𝒫 On
122114, 121jctil 558 . . . . . . . . . 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)))
123122ralrimiva 2949 . . . . . . . . 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)))
124 eqid 2610 . . . . . . . . . 10 OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) = OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒)))
125 eqid 2610 . . . . . . . . . 10 OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))) = OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒)))
126124, 125hsmexlem3 9133 . . . . . . . . 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))))
12780, 82, 123, 126syl21anc 1317 . . . . . . . 8 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → dom OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))) ∈ (har‘𝒫 (𝑋 × (𝐻𝑒))))
12879, 127syl5eqel 2692 . . . . . . 7 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (har‘𝒫 (𝑋 × (𝐻𝑒))))
12965hsmexlem8 9129 . . . . . . . 8 (𝑒 ∈ ω → (𝐻‘suc 𝑒) = (har‘𝒫 (𝑋 × (𝐻𝑒))))
130129ad2antrl 760 . . . . . . 7 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → (𝐻‘suc 𝑒) = (har‘𝒫 (𝑋 × (𝐻𝑒))))
131128, 130eleqtrrd 2691 . . . . . 6 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒))
132131expr 641 . . . . 5 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ 𝑒 ∈ ω) → (𝑑𝑆 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
13371, 132ralrimi 2940 . . . 4 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ 𝑒 ∈ ω) → ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒))
134133expcom 450 . . 3 (𝑒 ∈ ω → (∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) → ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
13510, 19, 28, 68, 134finds1 6987 . 2 (𝑐 ∈ ω → ∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐))
136135r19.21bi 2916 1 ((𝑐 ∈ ω ∧ 𝑑𝑆) → dom 𝑂 ∈ (𝐻𝑐))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   E cep 4947   × cxp 5036  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Oncon0 5640  suc csuc 5642  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  ωcom 6957  reccrdg 7392   ≼ cdom 7839  OrdIsocoi 8297  harchar 8344   ≼* cwdom 8345  TCctc 8495  𝑅1cr1 8508  rankcrnk 8509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-smo 7330  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-oi 8298  df-har 8346  df-wdom 8347  df-tc 8496  df-r1 8510  df-rank 8511 This theorem is referenced by:  hsmexlem5  9135
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