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Theorem hsmexlem4 9504
Description: Lemma for hsmex 9507. 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 6375 . . . . . . . . 9 (𝑐 = ∅ → ((𝑈𝑑)‘𝑐) = ((𝑈𝑑)‘∅))
32imaeq2d 5648 . . . . . . . 8 (𝑐 = ∅ → (rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘∅)))
4 oieq2 8625 . . . . . . . 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 2811 . . . . . 6 (𝑐 = ∅ → 𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘∅))))
76dmeqd 5494 . . . . 5 (𝑐 = ∅ → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))))
8 fveq2 6375 . . . . 5 (𝑐 = ∅ → (𝐻𝑐) = (𝐻‘∅))
97, 8eleq12d 2838 . . . 4 (𝑐 = ∅ → (dom 𝑂 ∈ (𝐻𝑐) ↔ dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅)))
109ralbidv 3133 . . 3 (𝑐 = ∅ → (∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐) ↔ ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅)))
11 fveq2 6375 . . . . . . . . 9 (𝑐 = 𝑒 → ((𝑈𝑑)‘𝑐) = ((𝑈𝑑)‘𝑒))
1211imaeq2d 5648 . . . . . . . 8 (𝑐 = 𝑒 → (rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘𝑒)))
13 oieq2 8625 . . . . . . . 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 2811 . . . . . 6 (𝑐 = 𝑒𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))))
1615dmeqd 5494 . . . . 5 (𝑐 = 𝑒 → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))))
17 fveq2 6375 . . . . 5 (𝑐 = 𝑒 → (𝐻𝑐) = (𝐻𝑒))
1816, 17eleq12d 2838 . . . 4 (𝑐 = 𝑒 → (dom 𝑂 ∈ (𝐻𝑐) ↔ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒)))
1918ralbidv 3133 . . 3 (𝑐 = 𝑒 → (∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐) ↔ ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒)))
20 fveq2 6375 . . . . . . . . 9 (𝑐 = suc 𝑒 → ((𝑈𝑑)‘𝑐) = ((𝑈𝑑)‘suc 𝑒))
2120imaeq2d 5648 . . . . . . . 8 (𝑐 = suc 𝑒 → (rank “ ((𝑈𝑑)‘𝑐)) = (rank “ ((𝑈𝑑)‘suc 𝑒)))
22 oieq2 8625 . . . . . . . 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 2811 . . . . . 6 (𝑐 = suc 𝑒𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))))
2524dmeqd 5494 . . . . 5 (𝑐 = suc 𝑒 → dom 𝑂 = dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))))
26 fveq2 6375 . . . . 5 (𝑐 = suc 𝑒 → (𝐻𝑐) = (𝐻‘suc 𝑒))
2725, 26eleq12d 2838 . . . 4 (𝑐 = suc 𝑒 → (dom 𝑂 ∈ (𝐻𝑐) ↔ dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
2827ralbidv 3133 . . 3 (𝑐 = suc 𝑒 → (∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐) ↔ ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
29 imassrn 5659 . . . . . . 7 (rank “ ((𝑈𝑑)‘∅)) ⊆ ran rank
30 rankf 8872 . . . . . . . 8 rank: (𝑅1 “ On)⟶On
31 frn 6229 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → ran rank ⊆ On)
3230, 31ax-mp 5 . . . . . . 7 ran rank ⊆ On
3329, 32sstri 3770 . . . . . 6 (rank “ ((𝑈𝑑)‘∅)) ⊆ On
34 vex 3353 . . . . . . . . 9 𝑑 ∈ V
35 hsmexlem4.u . . . . . . . . . 10 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
3635ituni0 9493 . . . . . . . . 9 (𝑑 ∈ V → ((𝑈𝑑)‘∅) = 𝑑)
3734, 36ax-mp 5 . . . . . . . 8 ((𝑈𝑑)‘∅) = 𝑑
3837imaeq2i 5646 . . . . . . 7 (rank “ ((𝑈𝑑)‘∅)) = (rank “ 𝑑)
39 ffun 6226 . . . . . . . . . 10 (rank: (𝑅1 “ On)⟶On → Fun rank)
4030, 39ax-mp 5 . . . . . . . . 9 Fun rank
41 wdomimag 8699 . . . . . . . . 9 ((Fun rank ∧ 𝑑 ∈ V) → (rank “ 𝑑) ≼* 𝑑)
4240, 34, 41mp2an 683 . . . . . . . 8 (rank “ 𝑑) ≼* 𝑑
43 sneq 4344 . . . . . . . . . . . . 13 (𝑎 = 𝑑 → {𝑎} = {𝑑})
4443fveq2d 6379 . . . . . . . . . . . 12 (𝑎 = 𝑑 → (TC‘{𝑎}) = (TC‘{𝑑}))
4544raleqdv 3292 . . . . . . . . . . 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 3523 . . . . . . . . . 10 (𝑑𝑆 ↔ (𝑑 (𝑅1 “ On) ∧ ∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋))
4847simprbi 490 . . . . . . . . 9 (𝑑𝑆 → ∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋)
49 snex 5064 . . . . . . . . . . . 12 {𝑑} ∈ V
50 tcid 8830 . . . . . . . . . . . 12 ({𝑑} ∈ V → {𝑑} ⊆ (TC‘{𝑑}))
5149, 50ax-mp 5 . . . . . . . . . . 11 {𝑑} ⊆ (TC‘{𝑑})
52 vsnid 4367 . . . . . . . . . . 11 𝑑 ∈ {𝑑}
5351, 52sselii 3758 . . . . . . . . . 10 𝑑 ∈ (TC‘{𝑑})
54 breq1 4812 . . . . . . . . . . 11 (𝑏 = 𝑑 → (𝑏𝑋𝑑𝑋))
5554rspcv 3457 . . . . . . . . . 10 (𝑑 ∈ (TC‘{𝑑}) → (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋𝑑𝑋))
5653, 55ax-mp 5 . . . . . . . . 9 (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋𝑑𝑋)
57 domwdom 8686 . . . . . . . . 9 (𝑑𝑋𝑑* 𝑋)
5848, 56, 573syl 18 . . . . . . . 8 (𝑑𝑆𝑑* 𝑋)
59 wdomtr 8687 . . . . . . . 8 (((rank “ 𝑑) ≼* 𝑑𝑑* 𝑋) → (rank “ 𝑑) ≼* 𝑋)
6042, 58, 59sylancr 581 . . . . . . 7 (𝑑𝑆 → (rank “ 𝑑) ≼* 𝑋)
6138, 60syl5eqbr 4844 . . . . . 6 (𝑑𝑆 → (rank “ ((𝑈𝑑)‘∅)) ≼* 𝑋)
62 eqid 2765 . . . . . . 7 OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) = OrdIso( E , (rank “ ((𝑈𝑑)‘∅)))
6362hsmexlem1 9501 . . . . . 6 (((rank “ ((𝑈𝑑)‘∅)) ⊆ On ∧ (rank “ ((𝑈𝑑)‘∅)) ≼* 𝑋) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (har‘𝒫 𝑋))
6433, 61, 63sylancr 581 . . . . 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 9498 . . . . 5 (𝐻‘∅) = (har‘𝒫 𝑋)
6764, 66syl6eleqr 2855 . . . 4 (𝑑𝑆 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅))
6867rgen 3069 . . 3 𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘∅))) ∈ (𝐻‘∅)
69 nfra1 3088 . . . . . 6 𝑑𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒)
70 nfv 2009 . . . . . 6 𝑑 𝑒 ∈ ω
7169, 70nfan 1998 . . . . 5 𝑑(∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ 𝑒 ∈ ω)
7235ituniiun 9497 . . . . . . . . . . . . 13 (𝑑 ∈ V → ((𝑈𝑑)‘suc 𝑒) = 𝑓𝑑 ((𝑈𝑓)‘𝑒))
7334, 72ax-mp 5 . . . . . . . . . . . 12 ((𝑈𝑑)‘suc 𝑒) = 𝑓𝑑 ((𝑈𝑓)‘𝑒)
7473imaeq2i 5646 . . . . . . . . . . 11 (rank “ ((𝑈𝑑)‘suc 𝑒)) = (rank “ 𝑓𝑑 ((𝑈𝑓)‘𝑒))
75 imaiun 6695 . . . . . . . . . . 11 (rank “ 𝑓𝑑 ((𝑈𝑓)‘𝑒)) = 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))
7674, 75eqtri 2787 . . . . . . . . . 10 (rank “ ((𝑈𝑑)‘suc 𝑒)) = 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))
77 oieq2 8625 . . . . . . . . . 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 5493 . . . . . . . 8 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) = dom OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒)))
8058ad2antll 720 . . . . . . . . 9 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → 𝑑* 𝑋)
8165hsmexlem9 9500 . . . . . . . . . 10 (𝑒 ∈ ω → (𝐻𝑒) ∈ On)
8281ad2antrl 719 . . . . . . . . 9 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → (𝐻𝑒) ∈ On)
83 fveq2 6375 . . . . . . . . . . . . . . . . 17 (𝑑 = 𝑓 → (𝑈𝑑) = (𝑈𝑓))
8483fveq1d 6377 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑓 → ((𝑈𝑑)‘𝑒) = ((𝑈𝑓)‘𝑒))
8584imaeq2d 5648 . . . . . . . . . . . . . . 15 (𝑑 = 𝑓 → (rank “ ((𝑈𝑑)‘𝑒)) = (rank “ ((𝑈𝑓)‘𝑒)))
86 oieq2 8625 . . . . . . . . . . . . . . 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 5494 . . . . . . . . . . . . 13 (𝑑 = 𝑓 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) = dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))))
8988eleq1d 2829 . . . . . . . . . . . 12 (𝑑 = 𝑓 → (dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ↔ dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) ∈ (𝐻𝑒)))
90 simpll 783 . . . . . . . . . . . 12 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒))
91 ssrab2 3847 . . . . . . . . . . . . . . . . . . 19 {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋} ⊆ (𝑅1 “ On)
9246, 91eqsstri 3795 . . . . . . . . . . . . . . . . . 18 𝑆 (𝑅1 “ On)
9392sseli 3757 . . . . . . . . . . . . . . . . 17 (𝑑𝑆𝑑 (𝑅1 “ On))
94 r1elssi 8883 . . . . . . . . . . . . . . . . 17 (𝑑 (𝑅1 “ On) → 𝑑 (𝑅1 “ On))
9593, 94syl 17 . . . . . . . . . . . . . . . 16 (𝑑𝑆𝑑 (𝑅1 “ On))
9695sselda 3761 . . . . . . . . . . . . . . 15 ((𝑑𝑆𝑓𝑑) → 𝑓 (𝑅1 “ On))
97 snssi 4493 . . . . . . . . . . . . . . . . . . 19 (𝑓𝑑 → {𝑓} ⊆ 𝑑)
9834tcss 8835 . . . . . . . . . . . . . . . . . . 19 ({𝑓} ⊆ 𝑑 → (TC‘{𝑓}) ⊆ (TC‘𝑑))
9997, 98syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓𝑑 → (TC‘{𝑓}) ⊆ (TC‘𝑑))
10049tcel 8836 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∈ {𝑑} → (TC‘𝑑) ⊆ (TC‘{𝑑}))
10152, 100mp1i 13 . . . . . . . . . . . . . . . . . 18 (𝑓𝑑 → (TC‘𝑑) ⊆ (TC‘{𝑑}))
10299, 101sstrd 3771 . . . . . . . . . . . . . . . . 17 (𝑓𝑑 → (TC‘{𝑓}) ⊆ (TC‘{𝑑}))
103 ssralv 3826 . . . . . . . . . . . . . . . . 17 ((TC‘{𝑓}) ⊆ (TC‘{𝑑}) → (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋 → ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
104102, 103syl 17 . . . . . . . . . . . . . . . 16 (𝑓𝑑 → (∀𝑏 ∈ (TC‘{𝑑})𝑏𝑋 → ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
10548, 104mpan9 502 . . . . . . . . . . . . . . 15 ((𝑑𝑆𝑓𝑑) → ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋)
106 sneq 4344 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑓 → {𝑎} = {𝑓})
107106fveq2d 6379 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑓 → (TC‘{𝑎}) = (TC‘{𝑓}))
108107raleqdv 3292 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑓 → (∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋 ↔ ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
109108, 46elrab2 3523 . . . . . . . . . . . . . . 15 (𝑓𝑆 ↔ (𝑓 (𝑅1 “ On) ∧ ∀𝑏 ∈ (TC‘{𝑓})𝑏𝑋))
11096, 105, 109sylanbrc 578 . . . . . . . . . . . . . 14 ((𝑑𝑆𝑓𝑑) → 𝑓𝑆)
111110adantll 705 . . . . . . . . . . . . 13 (((𝑒 ∈ ω ∧ 𝑑𝑆) ∧ 𝑓𝑑) → 𝑓𝑆)
112111adantll 705 . . . . . . . . . . . 12 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → 𝑓𝑆)
11389, 90, 112rspcdva 3467 . . . . . . . . . . 11 (((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) ∧ 𝑓𝑑) → dom OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) ∈ (𝐻𝑒))
114 imassrn 5659 . . . . . . . . . . . . 13 (rank “ ((𝑈𝑓)‘𝑒)) ⊆ ran rank
115114, 32sstri 3770 . . . . . . . . . . . 12 (rank “ ((𝑈𝑓)‘𝑒)) ⊆ On
116 fvex 6388 . . . . . . . . . . . . . . 15 ((𝑈𝑓)‘𝑒) ∈ V
117116funimaex 6154 . . . . . . . . . . . . . 14 (Fun rank → (rank “ ((𝑈𝑓)‘𝑒)) ∈ V)
11840, 117ax-mp 5 . . . . . . . . . . . . 13 (rank “ ((𝑈𝑓)‘𝑒)) ∈ V
119118elpw 4321 . . . . . . . . . . . 12 ((rank “ ((𝑈𝑓)‘𝑒)) ∈ 𝒫 On ↔ (rank “ ((𝑈𝑓)‘𝑒)) ⊆ On)
120115, 119mpbir 222 . . . . . . . . . . 11 (rank “ ((𝑈𝑓)‘𝑒)) ∈ 𝒫 On
121113, 120jctil 515 . . . . . . . . . 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)))
122121ralrimiva 3113 . . . . . . . . 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)))
123 eqid 2765 . . . . . . . . . 10 OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒))) = OrdIso( E , (rank “ ((𝑈𝑓)‘𝑒)))
124 eqid 2765 . . . . . . . . . 10 OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))) = OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒)))
125123, 124hsmexlem3 9503 . . . . . . . . 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))))
12680, 82, 122, 125syl21anc 866 . . . . . . . 8 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → dom OrdIso( E , 𝑓𝑑 (rank “ ((𝑈𝑓)‘𝑒))) ∈ (har‘𝒫 (𝑋 × (𝐻𝑒))))
12779, 126syl5eqel 2848 . . . . . . 7 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (har‘𝒫 (𝑋 × (𝐻𝑒))))
12865hsmexlem8 9499 . . . . . . . 8 (𝑒 ∈ ω → (𝐻‘suc 𝑒) = (har‘𝒫 (𝑋 × (𝐻𝑒))))
129128ad2antrl 719 . . . . . . 7 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → (𝐻‘suc 𝑒) = (har‘𝒫 (𝑋 × (𝐻𝑒))))
130127, 129eleqtrrd 2847 . . . . . 6 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ (𝑒 ∈ ω ∧ 𝑑𝑆)) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒))
131130expr 448 . . . . 5 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ 𝑒 ∈ ω) → (𝑑𝑆 → dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
13271, 131ralrimi 3104 . . . 4 ((∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) ∧ 𝑒 ∈ ω) → ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒))
133132expcom 402 . . 3 (𝑒 ∈ ω → (∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑒))) ∈ (𝐻𝑒) → ∀𝑑𝑆 dom OrdIso( E , (rank “ ((𝑈𝑑)‘suc 𝑒))) ∈ (𝐻‘suc 𝑒)))
13410, 19, 28, 68, 133finds1 7293 . 2 (𝑐 ∈ ω → ∀𝑑𝑆 dom 𝑂 ∈ (𝐻𝑐))
135134r19.21bi 3079 1 ((𝑐 ∈ ω ∧ 𝑑𝑆) → dom 𝑂 ∈ (𝐻𝑐))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334   cuni 4594   ciun 4676   class class class wbr 4809  cmpt 4888   E cep 5189   × cxp 5275  dom cdm 5277  ran crn 5278  cres 5279  cima 5280  Oncon0 5908  suc csuc 5910  Fun wfun 6062  wf 6064  cfv 6068  ωcom 7263  reccrdg 7709  cdom 8158  OrdIsocoi 8621  harchar 8668  * cwdom 8669  TCctc 8827  𝑅1cr1 8840  rankcrnk 8841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-smo 7647  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-oi 8622  df-har 8670  df-wdom 8671  df-tc 8828  df-r1 8842  df-rank 8843
This theorem is referenced by:  hsmexlem5  9505
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