MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hsmexlem5 Structured version   Visualization version   GIF version

Theorem hsmexlem5 9464
Description: Lemma for hsmex 9466. Combining the above constraints, along with itunitc 9455 and tcrank 8922, 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 3828 . . . . . . . 8 {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋} ⊆ (𝑅1 “ On)
31, 2eqsstri 3776 . . . . . . 7 𝑆 (𝑅1 “ On)
43sseli 3740 . . . . . 6 (𝑑𝑆𝑑 (𝑅1 “ On))
5 tcrank 8922 . . . . . 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 9451 . . . . . . . 8 (𝑑𝑆 → (𝑈𝑑) Fn ω)
9 fniunfv 6669 . . . . . . . 8 ((𝑈𝑑) Fn ω → 𝑐 ∈ ω ((𝑈𝑑)‘𝑐) = ran (𝑈𝑑))
108, 9syl 17 . . . . . . 7 (𝑑𝑆 𝑐 ∈ ω ((𝑈𝑑)‘𝑐) = ran (𝑈𝑑))
117itunitc 9455 . . . . . . 7 (TC‘𝑑) = ran (𝑈𝑑)
1210, 11syl6reqr 2813 . . . . . 6 (𝑑𝑆 → (TC‘𝑑) = 𝑐 ∈ ω ((𝑈𝑑)‘𝑐))
1312imaeq2d 5624 . . . . 5 (𝑑𝑆 → (rank “ (TC‘𝑑)) = (rank “ 𝑐 ∈ ω ((𝑈𝑑)‘𝑐)))
14 imaiun 6667 . . . . . 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 2798 . . . 4 (𝑑𝑆 → (rank‘𝑑) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
17 dmresi 5615 . . . 4 dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))
1816, 17syl6eqr 2812 . . 3 (𝑑𝑆 → (rank‘𝑑) = dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
19 rankon 8833 . . . . . 6 (rank‘𝑑) ∈ On
2016, 19syl6eqelr 2848 . . . . 5 (𝑑𝑆 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) ∈ On)
21 eloni 5894 . . . . 5 ( 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) ∈ On → Ord 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
22 oiid 8613 . . . . 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 5481 . . 3 (𝑑𝑆 → dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
2518, 24eqtr4d 2797 . 2 (𝑑𝑆 → (rank‘𝑑) = dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
26 omex 8715 . . . 4 ω ∈ V
27 wdomref 8644 . . . 4 (ω ∈ V → ω ≼* ω)
2826, 27mp1i 13 . . 3 (𝑑𝑆 → ω ≼* ω)
29 frfnom 7700 . . . . . . 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 6146 . . . . . . 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 6669 . . . . . 6 (𝐻 Fn ω → 𝑎 ∈ ω (𝐻𝑎) = ran 𝐻)
3432, 33ax-mp 5 . . . . 5 𝑎 ∈ ω (𝐻𝑎) = ran 𝐻
35 iunon 7606 . . . . . . 7 ((ω ∈ V ∧ ∀𝑎 ∈ ω (𝐻𝑎) ∈ On) → 𝑎 ∈ ω (𝐻𝑎) ∈ On)
3626, 35mpan 708 . . . . . 6 (∀𝑎 ∈ ω (𝐻𝑎) ∈ On → 𝑎 ∈ ω (𝐻𝑎) ∈ On)
3730hsmexlem9 9459 . . . . . 6 (𝑎 ∈ ω → (𝐻𝑎) ∈ On)
3836, 37mprg 3064 . . . . 5 𝑎 ∈ ω (𝐻𝑎) ∈ On
3934, 38eqeltrri 2836 . . . 4 ran 𝐻 ∈ On
4039a1i 11 . . 3 (𝑑𝑆 ran 𝐻 ∈ On)
41 fvssunirn 6379 . . . . . 6 (𝐻𝑐) ⊆ ran 𝐻
42 hsmexlem4.x . . . . . . . 8 𝑋 ∈ V
43 eqid 2760 . . . . . . . 8 OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))
4442, 30, 7, 1, 43hsmexlem4 9463 . . . . . . 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 3742 . . . . 5 ((𝑑𝑆𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻)
47 imassrn 5635 . . . . . . 7 (rank “ ((𝑈𝑑)‘𝑐)) ⊆ ran rank
48 rankf 8832 . . . . . . . 8 rank: (𝑅1 “ On)⟶On
49 frn 6214 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → ran rank ⊆ On)
5048, 49ax-mp 5 . . . . . . 7 ran rank ⊆ On
5147, 50sstri 3753 . . . . . 6 (rank “ ((𝑈𝑑)‘𝑐)) ⊆ On
52 ffun 6209 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → Fun rank)
53 fvex 6363 . . . . . . . . 9 ((𝑈𝑑)‘𝑐) ∈ V
5453funimaex 6137 . . . . . . . 8 (Fun rank → (rank “ ((𝑈𝑑)‘𝑐)) ∈ V)
5548, 52, 54mp2b 10 . . . . . . 7 (rank “ ((𝑈𝑑)‘𝑐)) ∈ V
5655elpw 4308 . . . . . 6 ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ↔ (rank “ ((𝑈𝑑)‘𝑐)) ⊆ On)
5751, 56mpbir 221 . . . . 5 (rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On
5846, 57jctil 561 . . . 4 ((𝑑𝑆𝑐 ∈ ω) → ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻))
5958ralrimiva 3104 . . 3 (𝑑𝑆 → ∀𝑐 ∈ ω ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻))
60 eqid 2760 . . . 4 OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
6143, 60hsmexlem3 9462 . . 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 1476 . 2 (𝑑𝑆 → dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) ∈ (har‘𝒫 (ω × ran 𝐻)))
6325, 62eqeltrd 2839 1 (𝑑𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  wss 3715  𝒫 cpw 4302  {csn 4321   cuni 4588   ciun 4672   class class class wbr 4804  cmpt 4881   I cid 5173   E cep 5178   × cxp 5264  dom cdm 5266  ran crn 5267  cres 5268  cima 5269  Ord word 5883  Oncon0 5884  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  ωcom 7231  reccrdg 7675  cdom 8121  OrdIsocoi 8581  harchar 8628  * cwdom 8629  TCctc 8787  𝑅1cr1 8800  rankcrnk 8801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-smo 7613  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-oi 8582  df-har 8630  df-wdom 8631  df-tc 8788  df-r1 8802  df-rank 8803
This theorem is referenced by:  hsmexlem6  9465
  Copyright terms: Public domain W3C validator