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Theorem hsmexlem6 10045
Description: Lemma for hsmex 10046. (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
hsmexlem6 𝑆 ∈ V
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 hsmexlem6
StepHypRef Expression
1 fvex 6730 . 2 (𝑅1‘(har‘𝒫 (ω × ran 𝐻))) ∈ V
2 hsmexlem4.x . . . . 5 𝑋 ∈ V
3 hsmexlem4.h . . . . 5 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
4 hsmexlem4.u . . . . 5 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
5 hsmexlem4.s . . . . 5 𝑆 = {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋}
6 hsmexlem4.o . . . . 5 𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))
72, 3, 4, 5, 6hsmexlem5 10044 . . . 4 (𝑑𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻)))
85ssrab3 3995 . . . . . 6 𝑆 (𝑅1 “ On)
98sseli 3896 . . . . 5 (𝑑𝑆𝑑 (𝑅1 “ On))
10 harcl 9175 . . . . . 6 (har‘𝒫 (ω × ran 𝐻)) ∈ On
11 r1fnon 9383 . . . . . . 7 𝑅1 Fn On
1211fndmi 6482 . . . . . 6 dom 𝑅1 = On
1310, 12eleqtrri 2837 . . . . 5 (har‘𝒫 (ω × ran 𝐻)) ∈ dom 𝑅1
14 rankr1ag 9418 . . . . 5 ((𝑑 (𝑅1 “ On) ∧ (har‘𝒫 (ω × ran 𝐻)) ∈ dom 𝑅1) → (𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ran 𝐻))) ↔ (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻))))
159, 13, 14sylancl 589 . . . 4 (𝑑𝑆 → (𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ran 𝐻))) ↔ (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻))))
167, 15mpbird 260 . . 3 (𝑑𝑆𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ran 𝐻))))
1716ssriv 3905 . 2 𝑆 ⊆ (𝑅1‘(har‘𝒫 (ω × ran 𝐻)))
181, 17ssexi 5215 1 𝑆 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1543  wcel 2110  wral 3061  {crab 3065  Vcvv 3408  𝒫 cpw 4513  {csn 4541   cuni 4819   class class class wbr 5053  cmpt 5135   E cep 5459   × cxp 5549  dom cdm 5551  ran crn 5552  cres 5553  cima 5554  Oncon0 6213  cfv 6380  ωcom 7644  reccrdg 8145  cdom 8624  OrdIsocoi 9125  harchar 9172  TCctc 9352  𝑅1cr1 9378  rankcrnk 9379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-smo 8083  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-oi 9126  df-har 9173  df-wdom 9181  df-tc 9353  df-r1 9380  df-rank 9381
This theorem is referenced by:  hsmex  10046
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