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Theorem hsmexlem5 8203
Description: Lemma for hsmex 8205. Combining the above constraints, along with itunitc 8194 and tcrank 7701, 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 3344 . . . . . . . 8  |-  { a  e.  U. ( R1
" On )  | 
A. b  e.  ( TC `  { a } ) b  ~<_  X }  C_  U. ( R1 " On )
31, 2eqsstri 3294 . . . . . . 7  |-  S  C_  U. ( R1 " On )
43sseli 3262 . . . . . 6  |-  ( d  e.  S  ->  d  e.  U. ( R1 " On ) )
5 tcrank 7701 . . . . . 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 8190 . . . . . . . 8  |-  ( d  e.  S  ->  ( U `  d )  Fn  om )
9 fniunfv 5894 . . . . . . . 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 8194 . . . . . . 7  |-  ( TC
`  d )  = 
U. ran  ( U `  d )
1210, 11syl6reqr 2417 . . . . . 6  |-  ( d  e.  S  ->  ( TC `  d )  = 
U_ c  e.  om  ( ( U `  d ) `  c
) )
1312imaeq2d 5115 . . . . 5  |-  ( d  e.  S  ->  ( rank " ( TC `  d ) )  =  ( rank " U_ c  e.  om  (
( U `  d
) `  c )
) )
14 imaiun 5892 . . . . . 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 2402 . . . 4  |-  ( d  e.  S  ->  ( rank `  d )  = 
U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) ) )
17 dmresi 5108 . . . 4  |-  dom  (  _I  |`  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )  =  U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) )
1816, 17syl6eqr 2416 . . 3  |-  ( d  e.  S  ->  ( rank `  d )  =  dom  (  _I  |`  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) ) )
19 rankon 7614 . . . . . 6  |-  ( rank `  d )  e.  On
2016, 19syl6eqelr 2455 . . . . 5  |-  ( d  e.  S  ->  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
)  e.  On )
21 eloni 4505 . . . . 5  |-  ( U_ c  e.  om  ( rank " ( ( U `
 d ) `  c ) )  e.  On  ->  Ord  U_ c  e.  om  ( rank " (
( U `  d
) `  c )
) )
22 oiid 7403 . . . . 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 4984 . . 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 2401 . 2  |-  ( d  e.  S  ->  ( rank `  d )  =  dom OrdIso (  _E  ,  U_ c  e.  om  ( rank " ( ( U `  d ) `
 c ) ) ) )
26 omex 7491 . . . 4  |-  om  e.  _V
27 wdomref 7433 . . . 4  |-  ( om  e.  _V  ->  om  ~<_*  om )
2826, 27mp1i 11 . . 3  |-  ( d  e.  S  ->  om  ~<_*  om )
29 frfnom 6589 . . . . . . 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 5443 . . . . . . 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 5894 . . . . . 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 5646 . . . . . . 7  |-  ( H `
 a )  e. 
_V
3626, 35iunonOLD 6498 . . . . . 6  |-  ( A. a  e.  om  ( H `  a )  e.  On  ->  U_ a  e. 
om  ( H `  a )  e.  On )
3730hsmexlem9 8198 . . . . . 6  |-  ( a  e.  om  ->  ( H `  a )  e.  On )
3836, 37mprg 2697 . . . . 5  |-  U_ a  e.  om  ( H `  a )  e.  On
3934, 38eqeltrri 2437 . . . 4  |-  U. ran  H  e.  On
4039a1i 10 . . 3  |-  ( d  e.  S  ->  U. ran  H  e.  On )
41 fvssunirn 5658 . . . . . 6  |-  ( H `
 c )  C_  U.
ran  H
42 hsmexlem4.x . . . . . . . 8  |-  X  e. 
_V
43 eqid 2366 . . . . . . . 8  |- OrdIso (  _E  ,  ( rank " (
( U `  d
) `  c )
) )  = OrdIso (  _E  ,  ( rank " (
( U `  d
) `  c )
) )
4442, 30, 7, 1, 43hsmexlem4 8202 . . . . . . 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 3264 . . . . 5  |-  ( ( d  e.  S  /\  c  e.  om )  ->  dom OrdIso (  _E  , 
( rank " ( ( U `  d ) `
 c ) ) )  e.  U. ran  H )
47 imassrn 5128 . . . . . . 7  |-  ( rank " ( ( U `
 d ) `  c ) )  C_  ran  rank
48 rankf 7613 . . . . . . . 8  |-  rank : U. ( R1 " On ) --> On
49 frn 5501 . . . . . . . 8  |-  ( rank
: U. ( R1
" On ) --> On 
->  ran  rank  C_  On )
5048, 49ax-mp 8 . . . . . . 7  |-  ran  rank  C_  On
5147, 50sstri 3274 . . . . . 6  |-  ( rank " ( ( U `
 d ) `  c ) )  C_  On
52 ffun 5497 . . . . . . . 8  |-  ( rank
: U. ( R1
" On ) --> On 
->  Fun  rank )
53 fvex 5646 . . . . . . . . 9  |-  ( ( U `  d ) `
 c )  e. 
_V
5453funimaex 5435 . . . . . . . 8  |-  ( Fun 
rank  ->  ( rank " (
( U `  d
) `  c )
)  e.  _V )
5548, 52, 54mp2b 9 . . . . . . 7  |-  ( rank " ( ( U `
 d ) `  c ) )  e. 
_V
5655elpw 3720 . . . . . 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 2711 . . 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 2366 . . . 4  |- OrdIso (  _E  ,  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )  = OrdIso (  _E  ,  U_ c  e. 
om  ( rank " (
( U `  d
) `  c )
) )
6143, 60hsmexlem3 8201 . . 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 1182 . 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 2440 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 1647    e. wcel 1715   A.wral 2628   {crab 2632   _Vcvv 2873    C_ wss 3238   ~Pcpw 3714   {csn 3729   U.cuni 3929   U_ciun 4007   class class class wbr 4125    e. cmpt 4179    _E cep 4406    _I cid 4407   Ord word 4494   Oncon0 4495   omcom 4759    X. cxp 4790   dom cdm 4792   ran crn 4793    |` cres 4794   "cima 4795   Fun wfun 5352    Fn wfn 5353   -->wf 5354   ` cfv 5358   reccrdg 6564    ~<_ cdom 7004  OrdIsocoi 7371  harchar 7417    ~<_* cwdom 7418   TCctc 7568   R1cr1 7581   rankcrnk 7582
This theorem is referenced by:  hsmexlem6  8204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-1st 6249  df-2nd 6250  df-riota 6446  df-smo 6505  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-oi 7372  df-har 7419  df-wdom 7420  df-tc 7569  df-r1 7583  df-rank 7584
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