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

Theorem hsmexlem5 8266
Description: Lemma for hsmex 8268. Combining the above constraints, along with itunitc 8257 and tcrank 7764, 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 3388 . . . . . . . 8  |-  { a  e.  U. ( R1
" On )  | 
A. b  e.  ( TC `  { a } ) b  ~<_  X }  C_  U. ( R1 " On )
31, 2eqsstri 3338 . . . . . . 7  |-  S  C_  U. ( R1 " On )
43sseli 3304 . . . . . 6  |-  ( d  e.  S  ->  d  e.  U. ( R1 " On ) )
5 tcrank 7764 . . . . . 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 8253 . . . . . . . 8  |-  ( d  e.  S  ->  ( U `  d )  Fn  om )
9 fniunfv 5953 . . . . . . . 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 8257 . . . . . . 7  |-  ( TC
`  d )  = 
U. ran  ( U `  d )
1210, 11syl6reqr 2455 . . . . . 6  |-  ( d  e.  S  ->  ( TC `  d )  = 
U_ c  e.  om  ( ( U `  d ) `  c
) )
1312imaeq2d 5162 . . . . 5  |-  ( d  e.  S  ->  ( rank " ( TC `  d ) )  =  ( rank " U_ c  e.  om  (
( U `  d
) `  c )
) )
14 imaiun 5951 . . . . . 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 2440 . . . 4  |-  ( d  e.  S  ->  ( rank `  d )  = 
U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) ) )
17 dmresi 5155 . . . 4  |-  dom  (  _I  |`  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )  =  U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) )
1816, 17syl6eqr 2454 . . 3  |-  ( d  e.  S  ->  ( rank `  d )  =  dom  (  _I  |`  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) ) )
19 rankon 7677 . . . . . 6  |-  ( rank `  d )  e.  On
2016, 19syl6eqelr 2493 . . . . 5  |-  ( d  e.  S  ->  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
)  e.  On )
21 eloni 4551 . . . . 5  |-  ( U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) )  e.  On  ->  Ord  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) )
22 oiid 7466 . . . . 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 5031 . . 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 2439 . 2  |-  ( d  e.  S  ->  ( rank `  d )  =  dom OrdIso (  _E  ,  U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) ) ) )
26 omex 7554 . . . 4  |-  om  e.  _V
27 wdomref 7496 . . . 4  |-  ( om  e.  _V  ->  om  ~<_*  om )
2826, 27mp1i 12 . . 3  |-  ( d  e.  S  ->  om  ~<_*  om )
29 frfnom 6651 . . . . . . 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 5498 . . . . . . 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 5953 . . . . . 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 5701 . . . . . . 7  |-  ( H `
 a )  e. 
_V
3626, 35iunonOLD 6560 . . . . . 6  |-  ( A. a  e.  om  ( H `  a )  e.  On  ->  U_ a  e. 
om  ( H `  a )  e.  On )
3730hsmexlem9 8261 . . . . . 6  |-  ( a  e.  om  ->  ( H `  a )  e.  On )
3836, 37mprg 2735 . . . . 5  |-  U_ a  e.  om  ( H `  a )  e.  On
3934, 38eqeltrri 2475 . . . 4  |-  U. ran  H  e.  On
4039a1i 11 . . 3  |-  ( d  e.  S  ->  U. ran  H  e.  On )
41 fvssunirn 5713 . . . . . 6  |-  ( H `
 c )  C_  U.
ran  H
42 hsmexlem4.x . . . . . . . 8  |-  X  e. 
_V
43 eqid 2404 . . . . . . . 8  |- OrdIso (  _E  ,  ( rank " (
( U `  d
) `  c )
) )  = OrdIso (  _E  ,  ( rank " (
( U `  d
) `  c )
) )
4442, 30, 7, 1, 43hsmexlem4 8265 . . . . . . 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 3306 . . . . 5  |-  ( ( d  e.  S  /\  c  e.  om )  ->  dom OrdIso (  _E  , 
( rank " ( ( U `  d ) `
 c ) ) )  e.  U. ran  H )
47 imassrn 5175 . . . . . . 7  |-  ( rank " ( ( U `
 d ) `  c ) )  C_  ran  rank
48 rankf 7676 . . . . . . . 8  |-  rank : U. ( R1 " On ) --> On
49 frn 5556 . . . . . . . 8  |-  ( rank
: U. ( R1
" On ) --> On 
->  ran  rank  C_  On )
5048, 49ax-mp 8 . . . . . . 7  |-  ran  rank  C_  On
5147, 50sstri 3317 . . . . . 6  |-  ( rank " ( ( U `
 d ) `  c ) )  C_  On
52 ffun 5552 . . . . . . . 8  |-  ( rank
: U. ( R1
" On ) --> On 
->  Fun  rank )
53 fvex 5701 . . . . . . . . 9  |-  ( ( U `  d ) `
 c )  e. 
_V
5453funimaex 5490 . . . . . . . 8  |-  ( Fun 
rank  ->  ( rank " (
( U `  d
) `  c )
)  e.  _V )
5548, 52, 54mp2b 10 . . . . . . 7  |-  ( rank " ( ( U `
 d ) `  c ) )  e. 
_V
5655elpw 3765 . . . . . 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 2749 . . 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 2404 . . . 4  |- OrdIso (  _E  ,  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )  = OrdIso (  _E  ,  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )
6143, 60hsmexlem3 8264 . . 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 2478 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 2666   {crab 2670   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   {csn 3774   U.cuni 3975   U_ciun 4053   class class class wbr 4172    e. cmpt 4226    _E cep 4452    _I cid 4453   Ord word 4540   Oncon0 4541   omcom 4804    X. cxp 4835   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413   reccrdg 6626    ~<_ cdom 7066  OrdIsocoi 7434  harchar 7480    ~<_* cwdom 7481   TCctc 7631   R1cr1 7644   rankcrnk 7645
This theorem is referenced by:  hsmexlem6  8267
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-1st 6308  df-2nd 6309  df-riota 6508  df-smo 6567  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-oi 7435  df-har 7482  df-wdom 7483  df-tc 7632  df-r1 7646  df-rank 7647
  Copyright terms: Public domain W3C validator