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Theorem hsmexlem6 8057
Description: Lemmr for hsmex 8058. (Contributed by Stefan O'Rear, 14-Feb-2015.)
Hypotheses
Ref Expression
hsmexlem4.x  |-  X  e. 
_V
hsmexlem4.h  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
hsmexlem4.u  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
hsmexlem4.s  |-  S  =  { a  e.  U. ( R1 " On )  |  A. b  e.  ( TC `  {
a } ) b  ~<_  X }
hsmexlem4.o  |-  O  = OrdIso
(  _E  ,  (
rank " ( ( U `
 d ) `  c ) ) )
Assertion
Ref Expression
hsmexlem6  |-  S  e. 
_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 5539 . 2  |-  ( R1
`  (har `  ~P ( om  X.  U. ran  H ) ) )  e. 
_V
2 hsmexlem4.x . . . . 5  |-  X  e. 
_V
3 hsmexlem4.h . . . . 5  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
4 hsmexlem4.u . . . . 5  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
5 hsmexlem4.s . . . . 5  |-  S  =  { a  e.  U. ( R1 " On )  |  A. b  e.  ( TC `  {
a } ) b  ~<_  X }
6 hsmexlem4.o . . . . 5  |-  O  = OrdIso
(  _E  ,  (
rank " ( ( U `
 d ) `  c ) ) )
72, 3, 4, 5, 6hsmexlem5 8056 . . . 4  |-  ( d  e.  S  ->  ( rank `  d )  e.  (har `  ~P ( om  X.  U. ran  H
) ) )
8 ssrab2 3258 . . . . . . 7  |-  { a  e.  U. ( R1
" On )  | 
A. b  e.  ( TC `  { a } ) b  ~<_  X }  C_  U. ( R1 " On )
95, 8eqsstri 3208 . . . . . 6  |-  S  C_  U. ( R1 " On )
109sseli 3176 . . . . 5  |-  ( d  e.  S  ->  d  e.  U. ( R1 " On ) )
11 harcl 7275 . . . . . 6  |-  (har `  ~P ( om  X.  U. ran  H ) )  e.  On
12 r1fnon 7439 . . . . . . 7  |-  R1  Fn  On
13 fndm 5343 . . . . . . 7  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1412, 13ax-mp 8 . . . . . 6  |-  dom  R1  =  On
1511, 14eleqtrri 2356 . . . . 5  |-  (har `  ~P ( om  X.  U. ran  H ) )  e. 
dom  R1
16 rankr1ag 7474 . . . . 5  |-  ( ( d  e.  U. ( R1 " On )  /\  (har `  ~P ( om 
X.  U. ran  H ) )  e.  dom  R1 )  ->  ( d  e.  ( R1 `  (har `  ~P ( om  X.  U.
ran  H ) ) )  <->  ( rank `  d
)  e.  (har `  ~P ( om  X.  U. ran  H ) ) ) )
1710, 15, 16sylancl 643 . . . 4  |-  ( d  e.  S  ->  (
d  e.  ( R1
`  (har `  ~P ( om  X.  U. ran  H ) ) )  <->  ( rank `  d )  e.  (har
`  ~P ( om 
X.  U. ran  H ) ) ) )
187, 17mpbird 223 . . 3  |-  ( d  e.  S  ->  d  e.  ( R1 `  (har `  ~P ( om  X.  U.
ran  H ) ) ) )
1918ssriv 3184 . 2  |-  S  C_  ( R1 `  (har `  ~P ( om  X.  U. ran  H ) ) )
201, 19ssexi 4159 1  |-  S  e. 
_V
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   ~Pcpw 3625   {csn 3640   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    _E cep 4303   Oncon0 4392   omcom 4656    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   ` cfv 5255   reccrdg 6422    ~<_ cdom 6861  OrdIsocoi 7224  harchar 7270   TCctc 7421   R1cr1 7434   rankcrnk 7435
This theorem is referenced by:  hsmex  8058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-1st 6122  df-2nd 6123  df-riota 6304  df-smo 6363  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-oi 7225  df-har 7272  df-wdom 7273  df-tc 7422  df-r1 7436  df-rank 7437
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