MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hsmexlem5 Unicode version

Theorem hsmexlem5 8056
Description: Lemma for hsmex 8058. Combining the above constraints, along with itunitc 8047 and tcrank 7554, gives an effective constraint on the rank of  S. (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
hsmexlem5  |-  ( d  e.  S  ->  ( rank `  d )  e.  (har `  ~P ( om  X.  U. ran  H
) ) )
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 hsmexlem5
StepHypRef Expression
1 hsmexlem4.s . . . . . . . 8  |-  S  =  { a  e.  U. ( R1 " On )  |  A. b  e.  ( TC `  {
a } ) b  ~<_  X }
2 ssrab2 3258 . . . . . . . 8  |-  { a  e.  U. ( R1
" On )  | 
A. b  e.  ( TC `  { a } ) b  ~<_  X }  C_  U. ( R1 " On )
31, 2eqsstri 3208 . . . . . . 7  |-  S  C_  U. ( R1 " On )
43sseli 3176 . . . . . 6  |-  ( d  e.  S  ->  d  e.  U. ( R1 " On ) )
5 tcrank 7554 . . . . . 6  |-  ( d  e.  U. ( R1
" On )  -> 
( rank `  d )  =  ( rank " ( TC `  d ) ) )
64, 5syl 15 . . . . 5  |-  ( d  e.  S  ->  ( rank `  d )  =  ( rank " ( TC `  d ) ) )
7 hsmexlem4.u . . . . . . . . 9  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
87itunifn 8043 . . . . . . . 8  |-  ( d  e.  S  ->  ( U `  d )  Fn  om )
9 fniunfv 5773 . . . . . . . 8  |-  ( ( U `  d )  Fn  om  ->  U_ c  e.  om  ( ( U `
 d ) `  c )  =  U. ran  ( U `  d
) )
108, 9syl 15 . . . . . . 7  |-  ( d  e.  S  ->  U_ c  e.  om  ( ( U `
 d ) `  c )  =  U. ran  ( U `  d
) )
117itunitc 8047 . . . . . . 7  |-  ( TC
`  d )  = 
U. ran  ( U `  d )
1210, 11syl6reqr 2334 . . . . . 6  |-  ( d  e.  S  ->  ( TC `  d )  = 
U_ c  e.  om  ( ( U `  d ) `  c
) )
1312imaeq2d 5012 . . . . 5  |-  ( d  e.  S  ->  ( rank " ( TC `  d ) )  =  ( rank " U_ c  e.  om  (
( U `  d
) `  c )
) )
14 imaiun 5771 . . . . . 6  |-  ( rank " U_ c  e.  om  ( ( U `  d ) `  c
) )  =  U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) )
1514a1i 10 . . . . 5  |-  ( d  e.  S  ->  ( rank " U_ c  e. 
om  ( ( U `
 d ) `  c ) )  = 
U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) ) )
166, 13, 153eqtrd 2319 . . . 4  |-  ( d  e.  S  ->  ( rank `  d )  = 
U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) ) )
17 dmresi 5005 . . . 4  |-  dom  (  _I  |`  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )  =  U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) )
1816, 17syl6eqr 2333 . . 3  |-  ( d  e.  S  ->  ( rank `  d )  =  dom  (  _I  |`  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) ) )
19 rankon 7467 . . . . . 6  |-  ( rank `  d )  e.  On
2016, 19syl6eqelr 2372 . . . . 5  |-  ( d  e.  S  ->  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
)  e.  On )
21 eloni 4402 . . . . 5  |-  ( U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) )  e.  On  ->  Ord  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) )
22 oiid 7256 . . . . 5  |-  ( Ord  U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) )  -> OrdIso (  _E  ,  U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) ) )  =  (  _I  |`  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) ) )
2320, 21, 223syl 18 . . . 4  |-  ( d  e.  S  -> OrdIso (  _E  ,  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )  =  (  _I  |`  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) ) )
2423dmeqd 4881 . . 3  |-  ( d  e.  S  ->  dom OrdIso (  _E  ,  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) )  =  dom  (  _I  |`  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) ) )
2518, 24eqtr4d 2318 . 2  |-  ( d  e.  S  ->  ( rank `  d )  =  dom OrdIso (  _E  ,  U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) ) ) )
26 omex 7344 . . . 4  |-  om  e.  _V
27 wdomref 7286 . . . 4  |-  ( om  e.  _V  ->  om  ~<_*  om )
2826, 27mp1i 11 . . 3  |-  ( d  e.  S  ->  om  ~<_*  om )
29 frfnom 6447 . . . . . . 7  |-  ( rec ( ( z  e. 
_V  |->  (har `  ~P ( X  X.  z
) ) ) ,  (har `  ~P X ) )  |`  om )  Fn  om
30 hsmexlem4.h . . . . . . . 8  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
3130fneq1i 5338 . . . . . . 7  |-  ( H  Fn  om  <->  ( rec ( ( z  e. 
_V  |->  (har `  ~P ( X  X.  z
) ) ) ,  (har `  ~P X ) )  |`  om )  Fn  om )
3229, 31mpbir 200 . . . . . 6  |-  H  Fn  om
33 fniunfv 5773 . . . . . 6  |-  ( H  Fn  om  ->  U_ a  e.  om  ( H `  a )  =  U. ran  H )
3432, 33ax-mp 8 . . . . 5  |-  U_ a  e.  om  ( H `  a )  =  U. ran  H
35 fvex 5539 . . . . . . 7  |-  ( H `
 a )  e. 
_V
3626, 35iunonOLD 6356 . . . . . 6  |-  ( A. a  e.  om  ( H `  a )  e.  On  ->  U_ a  e. 
om  ( H `  a )  e.  On )
3730hsmexlem9 8051 . . . . . 6  |-  ( a  e.  om  ->  ( H `  a )  e.  On )
3836, 37mprg 2612 . . . . 5  |-  U_ a  e.  om  ( H `  a )  e.  On
3934, 38eqeltrri 2354 . . . 4  |-  U. ran  H  e.  On
4039a1i 10 . . 3  |-  ( d  e.  S  ->  U. ran  H  e.  On )
41 fvssunirn 5551 . . . . . 6  |-  ( H `
 c )  C_  U.
ran  H
42 hsmexlem4.x . . . . . . . 8  |-  X  e. 
_V
43 eqid 2283 . . . . . . . 8  |- OrdIso (  _E  ,  ( rank " (
( U `  d
) `  c )
) )  = OrdIso (  _E  ,  ( rank " (
( U `  d
) `  c )
) )
4442, 30, 7, 1, 43hsmexlem4 8055 . . . . . . 7  |-  ( ( c  e.  om  /\  d  e.  S )  ->  dom OrdIso (  _E  , 
( rank " ( ( U `  d ) `
 c ) ) )  e.  ( H `
 c ) )
4544ancoms 439 . . . . . 6  |-  ( ( d  e.  S  /\  c  e.  om )  ->  dom OrdIso (  _E  , 
( rank " ( ( U `  d ) `
 c ) ) )  e.  ( H `
 c ) )
4641, 45sseldi 3178 . . . . 5  |-  ( ( d  e.  S  /\  c  e.  om )  ->  dom OrdIso (  _E  , 
( rank " ( ( U `  d ) `
 c ) ) )  e.  U. ran  H )
47 imassrn 5025 . . . . . . 7  |-  ( rank " ( ( U `
 d ) `  c ) )  C_  ran  rank
48 rankf 7466 . . . . . . . 8  |-  rank : U. ( R1 " On ) --> On
49 frn 5395 . . . . . . . 8  |-  ( rank
: U. ( R1
" On ) --> On 
->  ran  rank  C_  On )
5048, 49ax-mp 8 . . . . . . 7  |-  ran  rank  C_  On
5147, 50sstri 3188 . . . . . 6  |-  ( rank " ( ( U `
 d ) `  c ) )  C_  On
52 ffun 5391 . . . . . . . 8  |-  ( rank
: U. ( R1
" On ) --> On 
->  Fun  rank )
53 fvex 5539 . . . . . . . . 9  |-  ( ( U `  d ) `
 c )  e. 
_V
5453funimaex 5330 . . . . . . . 8  |-  ( Fun 
rank  ->  ( rank " (
( U `  d
) `  c )
)  e.  _V )
5548, 52, 54mp2b 9 . . . . . . 7  |-  ( rank " ( ( U `
 d ) `  c ) )  e. 
_V
5655elpw 3631 . . . . . 6  |-  ( (
rank " ( ( U `
 d ) `  c ) )  e. 
~P On  <->  ( rank " ( ( U `  d ) `  c
) )  C_  On )
5751, 56mpbir 200 . . . . 5  |-  ( rank " ( ( U `
 d ) `  c ) )  e. 
~P On
5846, 57jctil 523 . . . 4  |-  ( ( d  e.  S  /\  c  e.  om )  ->  ( ( rank " (
( U `  d
) `  c )
)  e.  ~P On  /\ 
dom OrdIso (  _E  ,  (
rank " ( ( U `
 d ) `  c ) ) )  e.  U. ran  H
) )
5958ralrimiva 2626 . . 3  |-  ( d  e.  S  ->  A. c  e.  om  ( ( rank " ( ( U `
 d ) `  c ) )  e. 
~P On  /\  dom OrdIso (  _E  ,  ( rank " ( ( U `
 d ) `  c ) ) )  e.  U. ran  H
) )
60 eqid 2283 . . . 4  |- OrdIso (  _E  ,  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )  = OrdIso (  _E  ,  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )
6143, 60hsmexlem3 8054 . . 3  |-  ( ( ( om  ~<_*  om  /\  U. ran  H  e.  On )  /\  A. c  e.  om  (
( rank " ( ( U `  d ) `
 c ) )  e.  ~P On  /\  dom OrdIso (  _E  ,  (
rank " ( ( U `
 d ) `  c ) ) )  e.  U. ran  H
) )  ->  dom OrdIso (  _E  ,  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) )  e.  (har
`  ~P ( om 
X.  U. ran  H ) ) )
6228, 40, 59, 61syl21anc 1181 . 2  |-  ( d  e.  S  ->  dom OrdIso (  _E  ,  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) )  e.  (har
`  ~P ( om 
X.  U. ran  H ) ) )
6325, 62eqeltrd 2357 1  |-  ( d  e.  S  ->  ( rank `  d )  e.  (har `  ~P ( om  X.  U. ran  H
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827   U_ciun 3905   class class class wbr 4023    e. cmpt 4077    _E cep 4303    _I cid 4304   Ord word 4391   Oncon0 4392   omcom 4656    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255   reccrdg 6422    ~<_ cdom 6861  OrdIsocoi 7224  harchar 7270    ~<_* cwdom 7271   TCctc 7421   R1cr1 7434   rankcrnk 7435
This theorem is referenced by:  hsmexlem6  8057
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
  Copyright terms: Public domain W3C validator