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Theorem hsmexlem5 8310
Description: Lemma for hsmex 8312. Combining the above constraints, along with itunitc 8301 and tcrank 7808, 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 3428 . . . . . . . 8  |-  { a  e.  U. ( R1
" On )  | 
A. b  e.  ( TC `  { a } ) b  ~<_  X }  C_  U. ( R1 " On )
31, 2eqsstri 3378 . . . . . . 7  |-  S  C_  U. ( R1 " On )
43sseli 3344 . . . . . 6  |-  ( d  e.  S  ->  d  e.  U. ( R1 " On ) )
5 tcrank 7808 . . . . . 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 8297 . . . . . . . 8  |-  ( d  e.  S  ->  ( U `  d )  Fn  om )
9 fniunfv 5994 . . . . . . . 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 8301 . . . . . . 7  |-  ( TC
`  d )  = 
U. ran  ( U `  d )
1210, 11syl6reqr 2487 . . . . . 6  |-  ( d  e.  S  ->  ( TC `  d )  = 
U_ c  e.  om  ( ( U `  d ) `  c
) )
1312imaeq2d 5203 . . . . 5  |-  ( d  e.  S  ->  ( rank " ( TC `  d ) )  =  ( rank " U_ c  e.  om  (
( U `  d
) `  c )
) )
14 imaiun 5992 . . . . . 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 2472 . . . 4  |-  ( d  e.  S  ->  ( rank `  d )  = 
U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) ) )
17 dmresi 5196 . . . 4  |-  dom  (  _I  |`  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )  =  U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) )
1816, 17syl6eqr 2486 . . 3  |-  ( d  e.  S  ->  ( rank `  d )  =  dom  (  _I  |`  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) ) )
19 rankon 7721 . . . . . 6  |-  ( rank `  d )  e.  On
2016, 19syl6eqelr 2525 . . . . 5  |-  ( d  e.  S  ->  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
)  e.  On )
21 eloni 4591 . . . . 5  |-  ( U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) )  e.  On  ->  Ord  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) )
22 oiid 7510 . . . . 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 5072 . . 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 2471 . 2  |-  ( d  e.  S  ->  ( rank `  d )  =  dom OrdIso (  _E  ,  U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) ) ) )
26 omex 7598 . . . 4  |-  om  e.  _V
27 wdomref 7540 . . . 4  |-  ( om  e.  _V  ->  om  ~<_*  om )
2826, 27mp1i 12 . . 3  |-  ( d  e.  S  ->  om  ~<_*  om )
29 frfnom 6692 . . . . . . 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 5539 . . . . . . 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 5994 . . . . . 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 5742 . . . . . . 7  |-  ( H `
 a )  e. 
_V
3626, 35iunonOLD 6601 . . . . . 6  |-  ( A. a  e.  om  ( H `  a )  e.  On  ->  U_ a  e. 
om  ( H `  a )  e.  On )
3730hsmexlem9 8305 . . . . . 6  |-  ( a  e.  om  ->  ( H `  a )  e.  On )
3836, 37mprg 2775 . . . . 5  |-  U_ a  e.  om  ( H `  a )  e.  On
3934, 38eqeltrri 2507 . . . 4  |-  U. ran  H  e.  On
4039a1i 11 . . 3  |-  ( d  e.  S  ->  U. ran  H  e.  On )
41 fvssunirn 5754 . . . . . 6  |-  ( H `
 c )  C_  U.
ran  H
42 hsmexlem4.x . . . . . . . 8  |-  X  e. 
_V
43 eqid 2436 . . . . . . . 8  |- OrdIso (  _E  ,  ( rank " (
( U `  d
) `  c )
) )  = OrdIso (  _E  ,  ( rank " (
( U `  d
) `  c )
) )
4442, 30, 7, 1, 43hsmexlem4 8309 . . . . . . 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 3346 . . . . 5  |-  ( ( d  e.  S  /\  c  e.  om )  ->  dom OrdIso (  _E  , 
( rank " ( ( U `  d ) `
 c ) ) )  e.  U. ran  H )
47 imassrn 5216 . . . . . . 7  |-  ( rank " ( ( U `
 d ) `  c ) )  C_  ran  rank
48 rankf 7720 . . . . . . . 8  |-  rank : U. ( R1 " On ) --> On
49 frn 5597 . . . . . . . 8  |-  ( rank
: U. ( R1
" On ) --> On 
->  ran  rank  C_  On )
5048, 49ax-mp 8 . . . . . . 7  |-  ran  rank  C_  On
5147, 50sstri 3357 . . . . . 6  |-  ( rank " ( ( U `
 d ) `  c ) )  C_  On
52 ffun 5593 . . . . . . . 8  |-  ( rank
: U. ( R1
" On ) --> On 
->  Fun  rank )
53 fvex 5742 . . . . . . . . 9  |-  ( ( U `  d ) `
 c )  e. 
_V
5453funimaex 5531 . . . . . . . 8  |-  ( Fun 
rank  ->  ( rank " (
( U `  d
) `  c )
)  e.  _V )
5548, 52, 54mp2b 10 . . . . . . 7  |-  ( rank " ( ( U `
 d ) `  c ) )  e. 
_V
5655elpw 3805 . . . . . 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 2789 . . 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 2436 . . . 4  |- OrdIso (  _E  ,  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )  = OrdIso (  _E  ,  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )
6143, 60hsmexlem3 8308 . . 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 2510 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 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956    C_ wss 3320   ~Pcpw 3799   {csn 3814   U.cuni 4015   U_ciun 4093   class class class wbr 4212    e. cmpt 4266    _E cep 4492    _I cid 4493   Ord word 4580   Oncon0 4581   omcom 4845    X. cxp 4876   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881   Fun wfun 5448    Fn wfn 5449   -->wf 5450   ` cfv 5454   reccrdg 6667    ~<_ cdom 7107  OrdIsocoi 7478  harchar 7524    ~<_* cwdom 7525   TCctc 7675   R1cr1 7688   rankcrnk 7689
This theorem is referenced by:  hsmexlem6  8311
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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