MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hsmexlem6 Unicode version

Theorem hsmexlem6 8073
Description: Lemmr for hsmex 8074. (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 5555 . 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 8072 . . . 4  |-  ( d  e.  S  ->  ( rank `  d )  e.  (har `  ~P ( om  X.  U. ran  H
) ) )
8 ssrab2 3271 . . . . . . 7  |-  { a  e.  U. ( R1
" On )  | 
A. b  e.  ( TC `  { a } ) b  ~<_  X }  C_  U. ( R1 " On )
95, 8eqsstri 3221 . . . . . 6  |-  S  C_  U. ( R1 " On )
109sseli 3189 . . . . 5  |-  ( d  e.  S  ->  d  e.  U. ( R1 " On ) )
11 harcl 7291 . . . . . 6  |-  (har `  ~P ( om  X.  U. ran  H ) )  e.  On
12 r1fnon 7455 . . . . . . 7  |-  R1  Fn  On
13 fndm 5359 . . . . . . 7  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1412, 13ax-mp 8 . . . . . 6  |-  dom  R1  =  On
1511, 14eleqtrri 2369 . . . . 5  |-  (har `  ~P ( om  X.  U. ran  H ) )  e. 
dom  R1
16 rankr1ag 7490 . . . . 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 3197 . 2  |-  S  C_  ( R1 `  (har `  ~P ( om  X.  U. ran  H ) ) )
201, 19ssexi 4175 1  |-  S  e. 
_V
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   ~Pcpw 3638   {csn 3653   U.cuni 3843   class class class wbr 4039    e. cmpt 4093    _E cep 4319   Oncon0 4408   omcom 4672    X. cxp 4703   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708    Fn wfn 5266   ` cfv 5271   reccrdg 6438    ~<_ cdom 6877  OrdIsocoi 7240  harchar 7286   TCctc 7437   R1cr1 7450   rankcrnk 7451
This theorem is referenced by:  hsmex  8074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-1st 6138  df-2nd 6139  df-riota 6320  df-smo 6379  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-oi 7241  df-har 7288  df-wdom 7289  df-tc 7438  df-r1 7452  df-rank 7453
  Copyright terms: Public domain W3C validator