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Theorem hsmexlem6 8311
Description: Lemmr for hsmex 8312. (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 5742 . 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 8310 . . . 4  |-  ( d  e.  S  ->  ( rank `  d )  e.  (har `  ~P ( om  X.  U. ran  H
) ) )
8 ssrab2 3428 . . . . . . 7  |-  { a  e.  U. ( R1
" On )  | 
A. b  e.  ( TC `  { a } ) b  ~<_  X }  C_  U. ( R1 " On )
95, 8eqsstri 3378 . . . . . 6  |-  S  C_  U. ( R1 " On )
109sseli 3344 . . . . 5  |-  ( d  e.  S  ->  d  e.  U. ( R1 " On ) )
11 harcl 7529 . . . . . 6  |-  (har `  ~P ( om  X.  U. ran  H ) )  e.  On
12 r1fnon 7693 . . . . . . 7  |-  R1  Fn  On
13 fndm 5544 . . . . . . 7  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1412, 13ax-mp 8 . . . . . 6  |-  dom  R1  =  On
1511, 14eleqtrri 2509 . . . . 5  |-  (har `  ~P ( om  X.  U. ran  H ) )  e. 
dom  R1
16 rankr1ag 7728 . . . . 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 644 . . . 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 224 . . 3  |-  ( d  e.  S  ->  d  e.  ( R1 `  (har `  ~P ( om  X.  U.
ran  H ) ) ) )
1918ssriv 3352 . 2  |-  S  C_  ( R1 `  (har `  ~P ( om  X.  U. ran  H ) ) )
201, 19ssexi 4348 1  |-  S  e. 
_V
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956   ~Pcpw 3799   {csn 3814   U.cuni 4015   class class class wbr 4212    e. cmpt 4266    _E cep 4492   Oncon0 4581   omcom 4845    X. cxp 4876   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881    Fn wfn 5449   ` cfv 5454   reccrdg 6667    ~<_ cdom 7107  OrdIsocoi 7478  harchar 7524   TCctc 7675   R1cr1 7688   rankcrnk 7689
This theorem is referenced by:  hsmex  8312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-1st 6349  df-2nd 6350  df-riota 6549  df-smo 6608  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-oi 7479  df-har 7526  df-wdom 7527  df-tc 7676  df-r1 7690  df-rank 7691
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