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Theorem hsmexlem5 9290
Description: Lemma for hsmex 9292. Combining the above constraints, along with itunitc 9281 and tcrank 8785, gives an effective constraint on the rank of 𝑆. (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
hsmexlem5 (𝑑𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻)))
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 hsmexlem5
StepHypRef Expression
1 hsmexlem4.s . . . . . . . 8 𝑆 = {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋}
2 ssrab2 3720 . . . . . . . 8 {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋} ⊆ (𝑅1 “ On)
31, 2eqsstri 3668 . . . . . . 7 𝑆 (𝑅1 “ On)
43sseli 3632 . . . . . 6 (𝑑𝑆𝑑 (𝑅1 “ On))
5 tcrank 8785 . . . . . 6 (𝑑 (𝑅1 “ On) → (rank‘𝑑) = (rank “ (TC‘𝑑)))
64, 5syl 17 . . . . 5 (𝑑𝑆 → (rank‘𝑑) = (rank “ (TC‘𝑑)))
7 hsmexlem4.u . . . . . . . . 9 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
87itunifn 9277 . . . . . . . 8 (𝑑𝑆 → (𝑈𝑑) Fn ω)
9 fniunfv 6545 . . . . . . . 8 ((𝑈𝑑) Fn ω → 𝑐 ∈ ω ((𝑈𝑑)‘𝑐) = ran (𝑈𝑑))
108, 9syl 17 . . . . . . 7 (𝑑𝑆 𝑐 ∈ ω ((𝑈𝑑)‘𝑐) = ran (𝑈𝑑))
117itunitc 9281 . . . . . . 7 (TC‘𝑑) = ran (𝑈𝑑)
1210, 11syl6reqr 2704 . . . . . 6 (𝑑𝑆 → (TC‘𝑑) = 𝑐 ∈ ω ((𝑈𝑑)‘𝑐))
1312imaeq2d 5501 . . . . 5 (𝑑𝑆 → (rank “ (TC‘𝑑)) = (rank “ 𝑐 ∈ ω ((𝑈𝑑)‘𝑐)))
14 imaiun 6543 . . . . . 6 (rank “ 𝑐 ∈ ω ((𝑈𝑑)‘𝑐)) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))
1514a1i 11 . . . . 5 (𝑑𝑆 → (rank “ 𝑐 ∈ ω ((𝑈𝑑)‘𝑐)) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
166, 13, 153eqtrd 2689 . . . 4 (𝑑𝑆 → (rank‘𝑑) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
17 dmresi 5492 . . . 4 dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))
1816, 17syl6eqr 2703 . . 3 (𝑑𝑆 → (rank‘𝑑) = dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
19 rankon 8696 . . . . . 6 (rank‘𝑑) ∈ On
2016, 19syl6eqelr 2739 . . . . 5 (𝑑𝑆 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) ∈ On)
21 eloni 5771 . . . . 5 ( 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) ∈ On → Ord 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
22 oiid 8487 . . . . 5 (Ord 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) → OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
2320, 21, 223syl 18 . . . 4 (𝑑𝑆 → OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
2423dmeqd 5358 . . 3 (𝑑𝑆 → dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
2518, 24eqtr4d 2688 . 2 (𝑑𝑆 → (rank‘𝑑) = dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
26 omex 8578 . . . 4 ω ∈ V
27 wdomref 8518 . . . 4 (ω ∈ V → ω ≼* ω)
2826, 27mp1i 13 . . 3 (𝑑𝑆 → ω ≼* ω)
29 frfnom 7575 . . . . . . 7 (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω
30 hsmexlem4.h . . . . . . . 8 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
3130fneq1i 6023 . . . . . . 7 (𝐻 Fn ω ↔ (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω)
3229, 31mpbir 221 . . . . . 6 𝐻 Fn ω
33 fniunfv 6545 . . . . . 6 (𝐻 Fn ω → 𝑎 ∈ ω (𝐻𝑎) = ran 𝐻)
3432, 33ax-mp 5 . . . . 5 𝑎 ∈ ω (𝐻𝑎) = ran 𝐻
35 iunon 7481 . . . . . . 7 ((ω ∈ V ∧ ∀𝑎 ∈ ω (𝐻𝑎) ∈ On) → 𝑎 ∈ ω (𝐻𝑎) ∈ On)
3626, 35mpan 706 . . . . . 6 (∀𝑎 ∈ ω (𝐻𝑎) ∈ On → 𝑎 ∈ ω (𝐻𝑎) ∈ On)
3730hsmexlem9 9285 . . . . . 6 (𝑎 ∈ ω → (𝐻𝑎) ∈ On)
3836, 37mprg 2955 . . . . 5 𝑎 ∈ ω (𝐻𝑎) ∈ On
3934, 38eqeltrri 2727 . . . 4 ran 𝐻 ∈ On
4039a1i 11 . . 3 (𝑑𝑆 ran 𝐻 ∈ On)
41 fvssunirn 6255 . . . . . 6 (𝐻𝑐) ⊆ ran 𝐻
42 hsmexlem4.x . . . . . . . 8 𝑋 ∈ V
43 eqid 2651 . . . . . . . 8 OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))
4442, 30, 7, 1, 43hsmexlem4 9289 . . . . . . 7 ((𝑐 ∈ ω ∧ 𝑑𝑆) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ (𝐻𝑐))
4544ancoms 468 . . . . . 6 ((𝑑𝑆𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ (𝐻𝑐))
4641, 45sseldi 3634 . . . . 5 ((𝑑𝑆𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻)
47 imassrn 5512 . . . . . . 7 (rank “ ((𝑈𝑑)‘𝑐)) ⊆ ran rank
48 rankf 8695 . . . . . . . 8 rank: (𝑅1 “ On)⟶On
49 frn 6091 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → ran rank ⊆ On)
5048, 49ax-mp 5 . . . . . . 7 ran rank ⊆ On
5147, 50sstri 3645 . . . . . 6 (rank “ ((𝑈𝑑)‘𝑐)) ⊆ On
52 ffun 6086 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → Fun rank)
53 fvex 6239 . . . . . . . . 9 ((𝑈𝑑)‘𝑐) ∈ V
5453funimaex 6014 . . . . . . . 8 (Fun rank → (rank “ ((𝑈𝑑)‘𝑐)) ∈ V)
5548, 52, 54mp2b 10 . . . . . . 7 (rank “ ((𝑈𝑑)‘𝑐)) ∈ V
5655elpw 4197 . . . . . 6 ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ↔ (rank “ ((𝑈𝑑)‘𝑐)) ⊆ On)
5751, 56mpbir 221 . . . . 5 (rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On
5846, 57jctil 559 . . . 4 ((𝑑𝑆𝑐 ∈ ω) → ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻))
5958ralrimiva 2995 . . 3 (𝑑𝑆 → ∀𝑐 ∈ ω ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻))
60 eqid 2651 . . . 4 OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
6143, 60hsmexlem3 9288 . . 3 (((ω ≼* ω ∧ ran 𝐻 ∈ On) ∧ ∀𝑐 ∈ ω ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻)) → dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) ∈ (har‘𝒫 (ω × ran 𝐻)))
6228, 40, 59, 61syl21anc 1365 . 2 (𝑑𝑆 → dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) ∈ (har‘𝒫 (ω × ran 𝐻)))
6325, 62eqeltrd 2730 1 (𝑑𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191  {csn 4210   cuni 4468   ciun 4552   class class class wbr 4685  cmpt 4762   I cid 5052   E cep 5057   × cxp 5141  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Ord word 5760  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  ωcom 7107  reccrdg 7550  cdom 7995  OrdIsocoi 8455  harchar 8502  * cwdom 8503  TCctc 8650  𝑅1cr1 8663  rankcrnk 8664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-smo 7488  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-oi 8456  df-har 8504  df-wdom 8505  df-tc 8651  df-r1 8665  df-rank 8666
This theorem is referenced by:  hsmexlem6  9291
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