MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hsmexlem2 Unicode version

Theorem hsmexlem2 8053
Description: Lemma for hsmex 8058. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 8197 but use of order types allows to canonically choose the sub-bijections, removing the choice requirement. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
hsmexlem.f  |-  F  = OrdIso
(  _E  ,  B
)
hsmexlem.g  |-  G  = OrdIso
(  _E  ,  U_ a  e.  A  B
)
Assertion
Ref Expression
hsmexlem2  |-  ( ( A  e.  _V  /\  C  e.  On  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  dom  G  e.  (har `  ~P ( A  X.  C
) ) )
Distinct variable groups:    A, a    C, a
Allowed substitution hints:    B( a)    F( a)    G( a)

Proof of Theorem hsmexlem2
Dummy variables  b 
c  d  e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpwi 3633 . . . . . 6  |-  ( B  e.  ~P On  ->  B 
C_  On )
21adantr 451 . . . . 5  |-  ( ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  B  C_  On )
32ralimi 2618 . . . 4  |-  ( A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  A. a  e.  A  B  C_  On )
4 iunss 3943 . . . 4  |-  ( U_ a  e.  A  B  C_  On  <->  A. a  e.  A  B  C_  On )
53, 4sylibr 203 . . 3  |-  ( A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  U_ a  e.  A  B  C_  On )
653ad2ant3 978 . 2  |-  ( ( A  e.  _V  /\  C  e.  On  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  U_ a  e.  A  B  C_  On )
7 xpexg 4800 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  On )  ->  ( A  X.  C
)  e.  _V )
873adant3 975 . . 3  |-  ( ( A  e.  _V  /\  C  e.  On  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  ( A  X.  C )  e. 
_V )
9 nfv 1605 . . . . . . . . 9  |-  F/ a  C  e.  On
10 nfra1 2593 . . . . . . . . 9  |-  F/ a A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C )
119, 10nfan 1771 . . . . . . . 8  |-  F/ a ( C  e.  On  /\ 
A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C ) )
12 rsp 2603 . . . . . . . . 9  |-  ( A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  ( a  e.  A  ->  ( B  e.  ~P On  /\  dom  F  e.  C ) ) )
13 onelss 4434 . . . . . . . . . . . . . 14  |-  ( C  e.  On  ->  ( dom  F  e.  C  ->  dom  F  C_  C )
)
1413imp 418 . . . . . . . . . . . . 13  |-  ( ( C  e.  On  /\  dom  F  e.  C )  ->  dom  F  C_  C
)
1514adantrl 696 . . . . . . . . . . . 12  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  dom  F  C_  C )
16153adant3 975 . . . . . . . . . . 11  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C )  /\  b  e.  B )  ->  dom  F 
C_  C )
17 hsmexlem.f . . . . . . . . . . . . . . . . . . 19  |-  F  = OrdIso
(  _E  ,  B
)
1817oismo 7255 . . . . . . . . . . . . . . . . . 18  |-  ( B 
C_  On  ->  ( Smo 
F  /\  ran  F  =  B ) )
191, 18syl 15 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  ~P On  ->  ( Smo  F  /\  ran  F  =  B ) )
2019ad2antrl 708 . . . . . . . . . . . . . . . 16  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  -> 
( Smo  F  /\  ran  F  =  B ) )
2120simprd 449 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  ran  F  =  B )
2217oif 7245 . . . . . . . . . . . . . . 15  |-  F : dom  F --> B
2321, 22jctil 523 . . . . . . . . . . . . . 14  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  -> 
( F : dom  F --> B  /\  ran  F  =  B ) )
24 dffo2 5455 . . . . . . . . . . . . . 14  |-  ( F : dom  F -onto-> B  <->  ( F : dom  F --> B  /\  ran  F  =  B ) )
2523, 24sylibr 203 . . . . . . . . . . . . 13  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  F : dom  F -onto-> B
)
26 dffo3 5675 . . . . . . . . . . . . . 14  |-  ( F : dom  F -onto-> B  <->  ( F : dom  F --> B  /\  A. b  e.  B  E. e  e. 
dom  F  b  =  ( F `  e ) ) )
2726simprbi 450 . . . . . . . . . . . . 13  |-  ( F : dom  F -onto-> B  ->  A. b  e.  B  E. e  e.  dom  F  b  =  ( F `
 e ) )
28 rsp 2603 . . . . . . . . . . . . 13  |-  ( A. b  e.  B  E. e  e.  dom  F  b  =  ( F `  e )  ->  (
b  e.  B  ->  E. e  e.  dom  F  b  =  ( F `
 e ) ) )
2925, 27, 283syl 18 . . . . . . . . . . . 12  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  -> 
( b  e.  B  ->  E. e  e.  dom  F  b  =  ( F `
 e ) ) )
30293impia 1148 . . . . . . . . . . 11  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C )  /\  b  e.  B )  ->  E. e  e.  dom  F  b  =  ( F `  e
) )
31 ssrexv 3238 . . . . . . . . . . 11  |-  ( dom 
F  C_  C  ->  ( E. e  e.  dom  F  b  =  ( F `
 e )  ->  E. e  e.  C  b  =  ( F `  e ) ) )
3216, 30, 31sylc 56 . . . . . . . . . 10  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C )  /\  b  e.  B )  ->  E. e  e.  C  b  =  ( F `  e ) )
33323exp 1150 . . . . . . . . 9  |-  ( C  e.  On  ->  (
( B  e.  ~P On  /\  dom  F  e.  C )  ->  (
b  e.  B  ->  E. e  e.  C  b  =  ( F `  e ) ) ) )
3412, 33sylan9r 639 . . . . . . . 8  |-  ( ( C  e.  On  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  (
a  e.  A  -> 
( b  e.  B  ->  E. e  e.  C  b  =  ( F `  e ) ) ) )
3511, 34reximdai 2651 . . . . . . 7  |-  ( ( C  e.  On  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  ( E. a  e.  A  b  e.  B  ->  E. a  e.  A  E. e  e.  C  b  =  ( F `  e ) ) )
36353adant1 973 . . . . . 6  |-  ( ( A  e.  _V  /\  C  e.  On  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  ( E. a  e.  A  b  e.  B  ->  E. a  e.  A  E. e  e.  C  b  =  ( F `  e ) ) )
37 nfv 1605 . . . . . . 7  |-  F/ d E. e  e.  C  b  =  ( F `  e )
38 nfcv 2419 . . . . . . . 8  |-  F/_ a C
39 nfcv 2419 . . . . . . . . . . 11  |-  F/_ a  _E
40 nfcsb1v 3113 . . . . . . . . . . 11  |-  F/_ a [_ d  /  a ]_ B
4139, 40nfoi 7229 . . . . . . . . . 10  |-  F/_ aOrdIso (  _E  ,  [_ d  /  a ]_ B
)
42 nfcv 2419 . . . . . . . . . 10  |-  F/_ a
e
4341, 42nffv 5532 . . . . . . . . 9  |-  F/_ a
(OrdIso (  _E  ,  [_ d  /  a ]_ B ) `  e
)
4443nfeq2 2430 . . . . . . . 8  |-  F/ a  b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e )
4538, 44nfrex 2598 . . . . . . 7  |-  F/ a E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e )
46 csbeq1a 3089 . . . . . . . . . . . 12  |-  ( a  =  d  ->  B  =  [_ d  /  a ]_ B )
47 oieq2 7228 . . . . . . . . . . . 12  |-  ( B  =  [_ d  / 
a ]_ B  -> OrdIso (  _E  ,  B )  = OrdIso
(  _E  ,  [_ d  /  a ]_ B
) )
4846, 47syl 15 . . . . . . . . . . 11  |-  ( a  =  d  -> OrdIso (  _E  ,  B )  = OrdIso
(  _E  ,  [_ d  /  a ]_ B
) )
4917, 48syl5eq 2327 . . . . . . . . . 10  |-  ( a  =  d  ->  F  = OrdIso (  _E  ,  [_ d  /  a ]_ B
) )
5049fveq1d 5527 . . . . . . . . 9  |-  ( a  =  d  ->  ( F `  e )  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B ) `  e
) )
5150eqeq2d 2294 . . . . . . . 8  |-  ( a  =  d  ->  (
b  =  ( F `
 e )  <->  b  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B
) `  e )
) )
5251rexbidv 2564 . . . . . . 7  |-  ( a  =  d  ->  ( E. e  e.  C  b  =  ( F `  e )  <->  E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B
) `  e )
) )
5337, 45, 52cbvrex 2761 . . . . . 6  |-  ( E. a  e.  A  E. e  e.  C  b  =  ( F `  e )  <->  E. d  e.  A  E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B
) `  e )
)
5436, 53syl6ib 217 . . . . 5  |-  ( ( A  e.  _V  /\  C  e.  On  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  ( E. a  e.  A  b  e.  B  ->  E. d  e.  A  E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B ) `  e
) ) )
55 eliun 3909 . . . . 5  |-  ( b  e.  U_ a  e.  A  B  <->  E. a  e.  A  b  e.  B )
56 vex 2791 . . . . . . . . . . 11  |-  d  e. 
_V
57 vex 2791 . . . . . . . . . . 11  |-  e  e. 
_V
5856, 57op1std 6130 . . . . . . . . . 10  |-  ( c  =  <. d ,  e
>.  ->  ( 1st `  c
)  =  d )
5958csbeq1d 3087 . . . . . . . . 9  |-  ( c  =  <. d ,  e
>.  ->  [_ ( 1st `  c
)  /  a ]_ B  =  [_ d  / 
a ]_ B )
60 oieq2 7228 . . . . . . . . 9  |-  ( [_ ( 1st `  c )  /  a ]_ B  =  [_ d  /  a ]_ B  -> OrdIso (  _E  ,  [_ ( 1st `  c )  /  a ]_ B )  = OrdIso (  _E  ,  [_ d  / 
a ]_ B ) )
6159, 60syl 15 . . . . . . . 8  |-  ( c  =  <. d ,  e
>.  -> OrdIso (  _E  ,  [_ ( 1st `  c )  /  a ]_ B
)  = OrdIso (  _E  ,  [_ d  /  a ]_ B ) )
6256, 57op2ndd 6131 . . . . . . . 8  |-  ( c  =  <. d ,  e
>.  ->  ( 2nd `  c
)  =  e )
6361, 62fveq12d 5531 . . . . . . 7  |-  ( c  =  <. d ,  e
>.  ->  (OrdIso (  _E  ,  [_ ( 1st `  c
)  /  a ]_ B ) `  ( 2nd `  c ) )  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e ) )
6463eqeq2d 2294 . . . . . 6  |-  ( c  =  <. d ,  e
>.  ->  ( b  =  (OrdIso (  _E  ,  [_ ( 1st `  c
)  /  a ]_ B ) `  ( 2nd `  c ) )  <-> 
b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e ) ) )
6564rexxp 4828 . . . . 5  |-  ( E. c  e.  ( A  X.  C ) b  =  (OrdIso (  _E  ,  [_ ( 1st `  c )  /  a ]_ B ) `  ( 2nd `  c ) )  <->  E. d  e.  A  E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e ) )
6654, 55, 653imtr4g 261 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  On  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  (
b  e.  U_ a  e.  A  B  ->  E. c  e.  ( A  X.  C ) b  =  (OrdIso (  _E  ,  [_ ( 1st `  c )  /  a ]_ B ) `  ( 2nd `  c ) ) ) )
6766imp 418 . . 3  |-  ( ( ( A  e.  _V  /\  C  e.  On  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  /\  b  e.  U_ a  e.  A  B )  ->  E. c  e.  ( A  X.  C
) b  =  (OrdIso (  _E  ,  [_ ( 1st `  c )  /  a ]_ B
) `  ( 2nd `  c ) ) )
688, 67wdomd 7295 . 2  |-  ( ( A  e.  _V  /\  C  e.  On  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  U_ a  e.  A  B  ~<_*  ( A  X.  C
) )
69 hsmexlem.g . . 3  |-  G  = OrdIso
(  _E  ,  U_ a  e.  A  B
)
7069hsmexlem1 8052 . 2  |-  ( (
U_ a  e.  A  B  C_  On  /\  U_ a  e.  A  B  ~<_*  ( A  X.  C ) )  ->  dom  G  e.  (har `  ~P ( A  X.  C ) ) )
716, 68, 70syl2anc 642 1  |-  ( ( A  e.  _V  /\  C  e.  On  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  dom  G  e.  (har `  ~P ( A  X.  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788   [_csb 3081    C_ wss 3152   ~Pcpw 3625   <.cop 3643   U_ciun 3905   class class class wbr 4023    _E cep 4303   Oncon0 4392    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251   -onto->wfo 5253   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   Smo wsmo 6362  OrdIsocoi 7224  harchar 7270    ~<_* cwdom 7271
This theorem is referenced by:  hsmexlem3  8054
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
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-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-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-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-oi 7225  df-har 7272  df-wdom 7273
  Copyright terms: Public domain W3C validator