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Theorem hsmexlem6 8814
Description: Lemma for hsmex 8815. (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 5866 . 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 8813 . . . 4  |-  ( d  e.  S  ->  ( rank `  d )  e.  (har `  ~P ( om  X.  U. ran  H
) ) )
8 ssrab2 3570 . . . . . . 7  |-  { a  e.  U. ( R1
" On )  | 
A. b  e.  ( TC `  { a } ) b  ~<_  X }  C_  U. ( R1 " On )
95, 8eqsstri 3519 . . . . . 6  |-  S  C_  U. ( R1 " On )
109sseli 3485 . . . . 5  |-  ( d  e.  S  ->  d  e.  U. ( R1 " On ) )
11 harcl 7990 . . . . . 6  |-  (har `  ~P ( om  X.  U. ran  H ) )  e.  On
12 r1fnon 8188 . . . . . . 7  |-  R1  Fn  On
13 fndm 5670 . . . . . . 7  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1412, 13ax-mp 5 . . . . . 6  |-  dom  R1  =  On
1511, 14eleqtrri 2530 . . . . 5  |-  (har `  ~P ( om  X.  U. ran  H ) )  e. 
dom  R1
16 rankr1ag 8223 . . . . 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 662 . . . 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 232 . . 3  |-  ( d  e.  S  ->  d  e.  ( R1 `  (har `  ~P ( om  X.  U.
ran  H ) ) ) )
1918ssriv 3493 . 2  |-  S  C_  ( R1 `  (har `  ~P ( om  X.  U. ran  H ) ) )
201, 19ssexi 4582 1  |-  S  e. 
_V
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1383    e. wcel 1804   A.wral 2793   {crab 2797   _Vcvv 3095   ~Pcpw 3997   {csn 4014   U.cuni 4234   class class class wbr 4437    |-> cmpt 4495    _E cep 4779   Oncon0 4868    X. cxp 4987   dom cdm 4989   ran crn 4990    |` cres 4991   "cima 4992    Fn wfn 5573   ` cfv 5578   omcom 6685   reccrdg 7077    ~<_ cdom 7516  OrdIsocoi 7937  harchar 7985   TCctc 8170   R1cr1 8183   rankcrnk 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-om 6686  df-1st 6785  df-2nd 6786  df-smo 7019  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-oi 7938  df-har 7987  df-wdom 7988  df-tc 8171  df-r1 8185  df-rank 8186
This theorem is referenced by:  hsmex  8815
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