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Theorem hsmexlem6 9651
 Description: Lemma for hsmex 9652. (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 6512 . 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 9650 . . . 4 (𝑑𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻)))
85ssrab3 3947 . . . . . 6 𝑆 (𝑅1 “ On)
98sseli 3854 . . . . 5 (𝑑𝑆𝑑 (𝑅1 “ On))
10 harcl 8820 . . . . . 6 (har‘𝒫 (ω × ran 𝐻)) ∈ On
11 r1fnon 8990 . . . . . . 7 𝑅1 Fn On
12 fndm 6288 . . . . . . 7 (𝑅1 Fn On → dom 𝑅1 = On)
1311, 12ax-mp 5 . . . . . 6 dom 𝑅1 = On
1410, 13eleqtrri 2865 . . . . 5 (har‘𝒫 (ω × ran 𝐻)) ∈ dom 𝑅1
15 rankr1ag 9025 . . . . 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))))
169, 14, 15sylancl 577 . . . 4 (𝑑𝑆 → (𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ran 𝐻))) ↔ (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻))))
177, 16mpbird 249 . . 3 (𝑑𝑆𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ran 𝐻))))
1817ssriv 3862 . 2 𝑆 ⊆ (𝑅1‘(har‘𝒫 (ω × ran 𝐻)))
191, 18ssexi 5082 1 𝑆 ∈ V
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1507   ∈ wcel 2050  ∀wral 3088  {crab 3092  Vcvv 3415  𝒫 cpw 4422  {csn 4441  ∪ cuni 4712   class class class wbr 4929   ↦ cmpt 5008   E cep 5316   × cxp 5405  dom cdm 5407  ran crn 5408   ↾ cres 5409   “ cima 5410  Oncon0 6029   Fn wfn 6183  ‘cfv 6188  ωcom 7396  reccrdg 7849   ≼ cdom 8304  OrdIsocoi 8768  harchar 8815  TCctc 8972  𝑅1cr1 8985  rankcrnk 8986 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-inf2 8898 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-smo 7787  df-recs 7812  df-rdg 7850  df-er 8089  df-en 8307  df-dom 8308  df-sdom 8309  df-oi 8769  df-har 8817  df-wdom 8818  df-tc 8973  df-r1 8987  df-rank 8988 This theorem is referenced by:  hsmex  9652
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