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

Theorem hsmexlem5 8274
Description: Lemma for hsmex 8276. Combining the above constraints, along with itunitc 8265 and tcrank 7772, 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 3396 . . . . . . . 8  |-  { a  e.  U. ( R1
" On )  | 
A. b  e.  ( TC `  { a } ) b  ~<_  X }  C_  U. ( R1 " On )
31, 2eqsstri 3346 . . . . . . 7  |-  S  C_  U. ( R1 " On )
43sseli 3312 . . . . . 6  |-  ( d  e.  S  ->  d  e.  U. ( R1 " On ) )
5 tcrank 7772 . . . . . 6  |-  ( d  e.  U. ( R1
" On )  -> 
( rank `  d )  =  ( rank " ( TC `  d ) ) )
64, 5syl 16 . . . . 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 8261 . . . . . . . 8  |-  ( d  e.  S  ->  ( U `  d )  Fn  om )
9 fniunfv 5961 . . . . . . . 8  |-  ( ( U `  d )  Fn  om  ->  U_ c  e.  om  ( ( U `
 d ) `  c )  =  U. ran  ( U `  d
) )
108, 9syl 16 . . . . . . 7  |-  ( d  e.  S  ->  U_ c  e.  om  ( ( U `
 d ) `  c )  =  U. ran  ( U `  d
) )
117itunitc 8265 . . . . . . 7  |-  ( TC
`  d )  = 
U. ran  ( U `  d )
1210, 11syl6reqr 2463 . . . . . 6  |-  ( d  e.  S  ->  ( TC `  d )  = 
U_ c  e.  om  ( ( U `  d ) `  c
) )
1312imaeq2d 5170 . . . . 5  |-  ( d  e.  S  ->  ( rank " ( TC `  d ) )  =  ( rank " U_ c  e.  om  (
( U `  d
) `  c )
) )
14 imaiun 5959 . . . . . 6  |-  ( rank " U_ c  e.  om  ( ( U `  d ) `  c
) )  =  U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) )
1514a1i 11 . . . . 5  |-  ( d  e.  S  ->  ( rank " U_ c  e. 
om  ( ( U `
 d ) `  c ) )  = 
U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) ) )
166, 13, 153eqtrd 2448 . . . 4  |-  ( d  e.  S  ->  ( rank `  d )  = 
U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) ) )
17 dmresi 5163 . . . 4  |-  dom  (  _I  |`  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )  =  U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) )
1816, 17syl6eqr 2462 . . 3  |-  ( d  e.  S  ->  ( rank `  d )  =  dom  (  _I  |`  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) ) )
19 rankon 7685 . . . . . 6  |-  ( rank `  d )  e.  On
2016, 19syl6eqelr 2501 . . . . 5  |-  ( d  e.  S  ->  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
)  e.  On )
21 eloni 4559 . . . . 5  |-  ( U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) )  e.  On  ->  Ord  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) )
22 oiid 7474 . . . . 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 19 . . . 4  |-  ( d  e.  S  -> OrdIso (  _E  ,  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )  =  (  _I  |`  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) ) )
2423dmeqd 5039 . . 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 2447 . 2  |-  ( d  e.  S  ->  ( rank `  d )  =  dom OrdIso (  _E  ,  U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) ) ) )
26 omex 7562 . . . 4  |-  om  e.  _V
27 wdomref 7504 . . . 4  |-  ( om  e.  _V  ->  om  ~<_*  om )
2826, 27mp1i 12 . . 3  |-  ( d  e.  S  ->  om  ~<_*  om )
29 frfnom 6659 . . . . . . 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 5506 . . . . . . 7  |-  ( H  Fn  om  <->  ( rec ( ( z  e. 
_V  |->  (har `  ~P ( X  X.  z
) ) ) ,  (har `  ~P X ) )  |`  om )  Fn  om )
3229, 31mpbir 201 . . . . . 6  |-  H  Fn  om
33 fniunfv 5961 . . . . . 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 5709 . . . . . . 7  |-  ( H `
 a )  e. 
_V
3626, 35iunonOLD 6568 . . . . . 6  |-  ( A. a  e.  om  ( H `  a )  e.  On  ->  U_ a  e. 
om  ( H `  a )  e.  On )
3730hsmexlem9 8269 . . . . . 6  |-  ( a  e.  om  ->  ( H `  a )  e.  On )
3836, 37mprg 2743 . . . . 5  |-  U_ a  e.  om  ( H `  a )  e.  On
3934, 38eqeltrri 2483 . . . 4  |-  U. ran  H  e.  On
4039a1i 11 . . 3  |-  ( d  e.  S  ->  U. ran  H  e.  On )
41 fvssunirn 5721 . . . . . 6  |-  ( H `
 c )  C_  U.
ran  H
42 hsmexlem4.x . . . . . . . 8  |-  X  e. 
_V
43 eqid 2412 . . . . . . . 8  |- OrdIso (  _E  ,  ( rank " (
( U `  d
) `  c )
) )  = OrdIso (  _E  ,  ( rank " (
( U `  d
) `  c )
) )
4442, 30, 7, 1, 43hsmexlem4 8273 . . . . . . 7  |-  ( ( c  e.  om  /\  d  e.  S )  ->  dom OrdIso (  _E  , 
( rank " ( ( U `  d ) `
 c ) ) )  e.  ( H `
 c ) )
4544ancoms 440 . . . . . 6  |-  ( ( d  e.  S  /\  c  e.  om )  ->  dom OrdIso (  _E  , 
( rank " ( ( U `  d ) `
 c ) ) )  e.  ( H `
 c ) )
4641, 45sseldi 3314 . . . . 5  |-  ( ( d  e.  S  /\  c  e.  om )  ->  dom OrdIso (  _E  , 
( rank " ( ( U `  d ) `
 c ) ) )  e.  U. ran  H )
47 imassrn 5183 . . . . . . 7  |-  ( rank " ( ( U `
 d ) `  c ) )  C_  ran  rank
48 rankf 7684 . . . . . . . 8  |-  rank : U. ( R1 " On ) --> On
49 frn 5564 . . . . . . . 8  |-  ( rank
: U. ( R1
" On ) --> On 
->  ran  rank  C_  On )
5048, 49ax-mp 8 . . . . . . 7  |-  ran  rank  C_  On
5147, 50sstri 3325 . . . . . 6  |-  ( rank " ( ( U `
 d ) `  c ) )  C_  On
52 ffun 5560 . . . . . . . 8  |-  ( rank
: U. ( R1
" On ) --> On 
->  Fun  rank )
53 fvex 5709 . . . . . . . . 9  |-  ( ( U `  d ) `
 c )  e. 
_V
5453funimaex 5498 . . . . . . . 8  |-  ( Fun 
rank  ->  ( rank " (
( U `  d
) `  c )
)  e.  _V )
5548, 52, 54mp2b 10 . . . . . . 7  |-  ( rank " ( ( U `
 d ) `  c ) )  e. 
_V
5655elpw 3773 . . . . . 6  |-  ( (
rank " ( ( U `
 d ) `  c ) )  e. 
~P On  <->  ( rank " ( ( U `  d ) `  c
) )  C_  On )
5751, 56mpbir 201 . . . . 5  |-  ( rank " ( ( U `
 d ) `  c ) )  e. 
~P On
5846, 57jctil 524 . . . 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 2757 . . 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 2412 . . . 4  |- OrdIso (  _E  ,  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )  = OrdIso (  _E  ,  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )
6143, 60hsmexlem3 8272 . . 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 1183 . 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 2486 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 359    = wceq 1649    e. wcel 1721   A.wral 2674   {crab 2678   _Vcvv 2924    C_ wss 3288   ~Pcpw 3767   {csn 3782   U.cuni 3983   U_ciun 4061   class class class wbr 4180    e. cmpt 4234    _E cep 4460    _I cid 4461   Ord word 4548   Oncon0 4549   omcom 4812    X. cxp 4843   dom cdm 4845   ran crn 4846    |` cres 4847   "cima 4848   Fun wfun 5415    Fn wfn 5416   -->wf 5417   ` cfv 5421   reccrdg 6634    ~<_ cdom 7074  OrdIsocoi 7442  harchar 7488    ~<_* cwdom 7489   TCctc 7639   R1cr1 7652   rankcrnk 7653
This theorem is referenced by:  hsmexlem6  8275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-1st 6316  df-2nd 6317  df-riota 6516  df-smo 6575  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-oi 7443  df-har 7490  df-wdom 7491  df-tc 7640  df-r1 7654  df-rank 7655
  Copyright terms: Public domain W3C validator