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Theorem hsmexlem5 9841
Description: Lemma for hsmex 9843. Combining the above constraints, along with itunitc 9832 and tcrank 9301, 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}
21ssrab3 4032 . . . . . . 7 𝑆 (𝑅1 “ On)
32sseli 3938 . . . . . 6 (𝑑𝑆𝑑 (𝑅1 “ On))
4 tcrank 9301 . . . . . 6 (𝑑 (𝑅1 “ On) → (rank‘𝑑) = (rank “ (TC‘𝑑)))
53, 4syl 17 . . . . 5 (𝑑𝑆 → (rank‘𝑑) = (rank “ (TC‘𝑑)))
6 hsmexlem4.u . . . . . . . . 9 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
76itunifn 9828 . . . . . . . 8 (𝑑𝑆 → (𝑈𝑑) Fn ω)
8 fniunfv 6989 . . . . . . . 8 ((𝑈𝑑) Fn ω → 𝑐 ∈ ω ((𝑈𝑑)‘𝑐) = ran (𝑈𝑑))
97, 8syl 17 . . . . . . 7 (𝑑𝑆 𝑐 ∈ ω ((𝑈𝑑)‘𝑐) = ran (𝑈𝑑))
106itunitc 9832 . . . . . . 7 (TC‘𝑑) = ran (𝑈𝑑)
119, 10syl6reqr 2876 . . . . . 6 (𝑑𝑆 → (TC‘𝑑) = 𝑐 ∈ ω ((𝑈𝑑)‘𝑐))
1211imaeq2d 5907 . . . . 5 (𝑑𝑆 → (rank “ (TC‘𝑑)) = (rank “ 𝑐 ∈ ω ((𝑈𝑑)‘𝑐)))
13 imaiun 6987 . . . . . 6 (rank “ 𝑐 ∈ ω ((𝑈𝑑)‘𝑐)) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))
1413a1i 11 . . . . 5 (𝑑𝑆 → (rank “ 𝑐 ∈ ω ((𝑈𝑑)‘𝑐)) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
155, 12, 143eqtrd 2861 . . . 4 (𝑑𝑆 → (rank‘𝑑) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
16 dmresi 5899 . . . 4 dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))
1715, 16eqtr4di 2875 . . 3 (𝑑𝑆 → (rank‘𝑑) = dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
18 rankon 9212 . . . . . 6 (rank‘𝑑) ∈ On
1915, 18eqeltrrdi 2923 . . . . 5 (𝑑𝑆 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) ∈ On)
20 eloni 6179 . . . . 5 ( 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) ∈ On → Ord 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
21 oiid 8993 . . . . 5 (Ord 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) → OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
2219, 20, 213syl 18 . . . 4 (𝑑𝑆 → OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
2322dmeqd 5751 . . 3 (𝑑𝑆 → dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
2417, 23eqtr4d 2860 . 2 (𝑑𝑆 → (rank‘𝑑) = dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
25 omex 9094 . . . 4 ω ∈ V
26 wdomref 9024 . . . 4 (ω ∈ V → ω ≼* ω)
2725, 26mp1i 13 . . 3 (𝑑𝑆 → ω ≼* ω)
28 frfnom 8057 . . . . . . 7 (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω
29 hsmexlem4.h . . . . . . . 8 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
3029fneq1i 6429 . . . . . . 7 (𝐻 Fn ω ↔ (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω)
3128, 30mpbir 234 . . . . . 6 𝐻 Fn ω
32 fniunfv 6989 . . . . . 6 (𝐻 Fn ω → 𝑎 ∈ ω (𝐻𝑎) = ran 𝐻)
3331, 32ax-mp 5 . . . . 5 𝑎 ∈ ω (𝐻𝑎) = ran 𝐻
34 iunon 7963 . . . . . . 7 ((ω ∈ V ∧ ∀𝑎 ∈ ω (𝐻𝑎) ∈ On) → 𝑎 ∈ ω (𝐻𝑎) ∈ On)
3525, 34mpan 689 . . . . . 6 (∀𝑎 ∈ ω (𝐻𝑎) ∈ On → 𝑎 ∈ ω (𝐻𝑎) ∈ On)
3629hsmexlem9 9836 . . . . . 6 (𝑎 ∈ ω → (𝐻𝑎) ∈ On)
3735, 36mprg 3144 . . . . 5 𝑎 ∈ ω (𝐻𝑎) ∈ On
3833, 37eqeltrri 2911 . . . 4 ran 𝐻 ∈ On
3938a1i 11 . . 3 (𝑑𝑆 ran 𝐻 ∈ On)
40 fvssunirn 6681 . . . . . 6 (𝐻𝑐) ⊆ ran 𝐻
41 hsmexlem4.x . . . . . . . 8 𝑋 ∈ V
42 eqid 2822 . . . . . . . 8 OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))
4341, 29, 6, 1, 42hsmexlem4 9840 . . . . . . 7 ((𝑐 ∈ ω ∧ 𝑑𝑆) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ (𝐻𝑐))
4443ancoms 462 . . . . . 6 ((𝑑𝑆𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ (𝐻𝑐))
4540, 44sseldi 3940 . . . . 5 ((𝑑𝑆𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻)
46 imassrn 5918 . . . . . . 7 (rank “ ((𝑈𝑑)‘𝑐)) ⊆ ran rank
47 rankf 9211 . . . . . . . 8 rank: (𝑅1 “ On)⟶On
48 frn 6500 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → ran rank ⊆ On)
4947, 48ax-mp 5 . . . . . . 7 ran rank ⊆ On
5046, 49sstri 3951 . . . . . 6 (rank “ ((𝑈𝑑)‘𝑐)) ⊆ On
51 ffun 6497 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → Fun rank)
52 fvex 6665 . . . . . . . . 9 ((𝑈𝑑)‘𝑐) ∈ V
5352funimaex 6420 . . . . . . . 8 (Fun rank → (rank “ ((𝑈𝑑)‘𝑐)) ∈ V)
5447, 51, 53mp2b 10 . . . . . . 7 (rank “ ((𝑈𝑑)‘𝑐)) ∈ V
5554elpw 4515 . . . . . 6 ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ↔ (rank “ ((𝑈𝑑)‘𝑐)) ⊆ On)
5650, 55mpbir 234 . . . . 5 (rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On
5745, 56jctil 523 . . . 4 ((𝑑𝑆𝑐 ∈ ω) → ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻))
5857ralrimiva 3174 . . 3 (𝑑𝑆 → ∀𝑐 ∈ ω ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻))
59 eqid 2822 . . . 4 OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
6042, 59hsmexlem3 9839 . . 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)))
6127, 39, 58, 60syl21anc 836 . 2 (𝑑𝑆 → dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) ∈ (har‘𝒫 (ω × ran 𝐻)))
6224, 61eqeltrd 2914 1 (𝑑𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  {crab 3134  Vcvv 3469  wss 3908  𝒫 cpw 4511  {csn 4539   cuni 4813   ciun 4894   class class class wbr 5042  cmpt 5122   I cid 5436   E cep 5441   × cxp 5530  dom cdm 5532  ran crn 5533  cres 5534  cima 5535  Ord word 6168  Oncon0 6169  Fun wfun 6328   Fn wfn 6329  wf 6330  cfv 6334  ωcom 7565  reccrdg 8032  cdom 8494  OrdIsocoi 8961  harchar 9008  * cwdom 9016  TCctc 9166  𝑅1cr1 9179  rankcrnk 9180
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-smo 7970  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-oi 8962  df-har 9009  df-wdom 9017  df-tc 9167  df-r1 9181  df-rank 9182
This theorem is referenced by:  hsmexlem6  9842
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