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Theorem hsmexlem2 8596
Description: Lemma for hsmex 8601. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 8739 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 3869 . . . . . 6  |-  ( B  e.  ~P On  ->  B 
C_  On )
21adantr 465 . . . . 5  |-  ( ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  B  C_  On )
32ralimi 2791 . . . 4  |-  ( A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  A. a  e.  A  B  C_  On )
4 iunss 4211 . . . 4  |-  ( U_ a  e.  A  B  C_  On  <->  A. a  e.  A  B  C_  On )
53, 4sylibr 212 . . 3  |-  ( A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  U_ a  e.  A  B  C_  On )
653ad2ant3 1011 . 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 6507 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  On )  ->  ( A  X.  C
)  e.  _V )
873adant3 1008 . . 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 1673 . . . . . . . . 9  |-  F/ a  C  e.  On
10 nfra1 2766 . . . . . . . . 9  |-  F/ a A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C )
119, 10nfan 1861 . . . . . . . 8  |-  F/ a ( C  e.  On  /\ 
A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C ) )
12 rsp 2776 . . . . . . . . 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 4761 . . . . . . . . . . . . . 14  |-  ( C  e.  On  ->  ( dom  F  e.  C  ->  dom  F  C_  C )
)
1413imp 429 . . . . . . . . . . . . 13  |-  ( ( C  e.  On  /\  dom  F  e.  C )  ->  dom  F  C_  C
)
1514adantrl 715 . . . . . . . . . . . 12  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  dom  F  C_  C )
16153adant3 1008 . . . . . . . . . . 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 7754 . . . . . . . . . . . . . . . . . 18  |-  ( B 
C_  On  ->  ( Smo 
F  /\  ran  F  =  B ) )
191, 18syl 16 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  ~P On  ->  ( Smo  F  /\  ran  F  =  B ) )
2019ad2antrl 727 . . . . . . . . . . . . . . . 16  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  -> 
( Smo  F  /\  ran  F  =  B ) )
2120simprd 463 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  ran  F  =  B )
2217oif 7744 . . . . . . . . . . . . . . 15  |-  F : dom  F --> B
2321, 22jctil 537 . . . . . . . . . . . . . 14  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  -> 
( F : dom  F --> B  /\  ran  F  =  B ) )
24 dffo2 5624 . . . . . . . . . . . . . 14  |-  ( F : dom  F -onto-> B  <->  ( F : dom  F --> B  /\  ran  F  =  B ) )
2523, 24sylibr 212 . . . . . . . . . . . . 13  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  F : dom  F -onto-> B
)
26 dffo3 5858 . . . . . . . . . . . . . 14  |-  ( F : dom  F -onto-> B  <->  ( F : dom  F --> B  /\  A. b  e.  B  E. e  e. 
dom  F  b  =  ( F `  e ) ) )
2726simprbi 464 . . . . . . . . . . . . 13  |-  ( F : dom  F -onto-> B  ->  A. b  e.  B  E. e  e.  dom  F  b  =  ( F `
 e ) )
28 rsp 2776 . . . . . . . . . . . . 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 20 . . . . . . . . . . . 12  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  -> 
( b  e.  B  ->  E. e  e.  dom  F  b  =  ( F `
 e ) ) )
30293impia 1184 . . . . . . . . . . 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 3417 . . . . . . . . . . 11  |-  ( dom 
F  C_  C  ->  ( E. e  e.  dom  F  b  =  ( F `
 e )  ->  E. e  e.  C  b  =  ( F `  e ) ) )
3216, 30, 31sylc 60 . . . . . . . . . 10  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C )  /\  b  e.  B )  ->  E. e  e.  C  b  =  ( F `  e ) )
33323exp 1186 . . . . . . . . 9  |-  ( C  e.  On  ->  (
( B  e.  ~P On  /\  dom  F  e.  C )  ->  (
b  e.  B  ->  E. e  e.  C  b  =  ( F `  e ) ) ) )
3412, 33sylan9r 658 . . . . . . . 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 2824 . . . . . . 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 1006 . . . . . 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 1673 . . . . . . 7  |-  F/ d E. e  e.  C  b  =  ( F `  e )
38 nfcv 2579 . . . . . . . 8  |-  F/_ a C
39 nfcv 2579 . . . . . . . . . . 11  |-  F/_ a  _E
40 nfcsb1v 3304 . . . . . . . . . . 11  |-  F/_ a [_ d  /  a ]_ B
4139, 40nfoi 7728 . . . . . . . . . 10  |-  F/_ aOrdIso (  _E  ,  [_ d  /  a ]_ B
)
42 nfcv 2579 . . . . . . . . . 10  |-  F/_ a
e
4341, 42nffv 5698 . . . . . . . . 9  |-  F/_ a
(OrdIso (  _E  ,  [_ d  /  a ]_ B ) `  e
)
4443nfeq2 2590 . . . . . . . 8  |-  F/ a  b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e )
4538, 44nfrex 2771 . . . . . . 7  |-  F/ a E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e )
46 csbeq1a 3297 . . . . . . . . . . . 12  |-  ( a  =  d  ->  B  =  [_ d  /  a ]_ B )
47 oieq2 7727 . . . . . . . . . . . 12  |-  ( B  =  [_ d  / 
a ]_ B  -> OrdIso (  _E  ,  B )  = OrdIso
(  _E  ,  [_ d  /  a ]_ B
) )
4846, 47syl 16 . . . . . . . . . . 11  |-  ( a  =  d  -> OrdIso (  _E  ,  B )  = OrdIso
(  _E  ,  [_ d  /  a ]_ B
) )
4917, 48syl5eq 2487 . . . . . . . . . 10  |-  ( a  =  d  ->  F  = OrdIso (  _E  ,  [_ d  /  a ]_ B
) )
5049fveq1d 5693 . . . . . . . . 9  |-  ( a  =  d  ->  ( F `  e )  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B ) `  e
) )
5150eqeq2d 2454 . . . . . . . 8  |-  ( a  =  d  ->  (
b  =  ( F `
 e )  <->  b  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B
) `  e )
) )
5251rexbidv 2736 . . . . . . 7  |-  ( a  =  d  ->  ( E. e  e.  C  b  =  ( F `  e )  <->  E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B
) `  e )
) )
5337, 45, 52cbvrex 2944 . . . . . 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 226 . . . . 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 4175 . . . . 5  |-  ( b  e.  U_ a  e.  A  B  <->  E. a  e.  A  b  e.  B )
56 vex 2975 . . . . . . . . . . 11  |-  d  e. 
_V
57 vex 2975 . . . . . . . . . . 11  |-  e  e. 
_V
5856, 57op1std 6587 . . . . . . . . . 10  |-  ( c  =  <. d ,  e
>.  ->  ( 1st `  c
)  =  d )
5958csbeq1d 3295 . . . . . . . . 9  |-  ( c  =  <. d ,  e
>.  ->  [_ ( 1st `  c
)  /  a ]_ B  =  [_ d  / 
a ]_ B )
60 oieq2 7727 . . . . . . . . 9  |-  ( [_ ( 1st `  c )  /  a ]_ B  =  [_ d  /  a ]_ B  -> OrdIso (  _E  ,  [_ ( 1st `  c )  /  a ]_ B )  = OrdIso (  _E  ,  [_ d  / 
a ]_ B ) )
6159, 60syl 16 . . . . . . . 8  |-  ( c  =  <. d ,  e
>.  -> OrdIso (  _E  ,  [_ ( 1st `  c )  /  a ]_ B
)  = OrdIso (  _E  ,  [_ d  /  a ]_ B ) )
6256, 57op2ndd 6588 . . . . . . . 8  |-  ( c  =  <. d ,  e
>.  ->  ( 2nd `  c
)  =  e )
6361, 62fveq12d 5697 . . . . . . 7  |-  ( c  =  <. d ,  e
>.  ->  (OrdIso (  _E  ,  [_ ( 1st `  c
)  /  a ]_ B ) `  ( 2nd `  c ) )  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e ) )
6463eqeq2d 2454 . . . . . 6  |-  ( c  =  <. d ,  e
>.  ->  ( b  =  (OrdIso (  _E  ,  [_ ( 1st `  c
)  /  a ]_ B ) `  ( 2nd `  c ) )  <-> 
b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e ) ) )
6564rexxp 4982 . . . . 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 270 . . . 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 429 . . 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 7796 . 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 8595 . 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 661 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   _Vcvv 2972   [_csb 3288    C_ wss 3328   ~Pcpw 3860   <.cop 3883   U_ciun 4171   class class class wbr 4292    _E cep 4630   Oncon0 4719    X. cxp 4838   dom cdm 4840   ran crn 4841   -->wf 5414   -onto->wfo 5416   ` cfv 5418   1stc1st 6575   2ndc2nd 6576   Smo wsmo 6806  OrdIsocoi 7723  harchar 7771    ~<_* cwdom 7772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-1st 6577  df-2nd 6578  df-smo 6807  df-recs 6832  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-oi 7724  df-har 7773  df-wdom 7774
This theorem is referenced by:  hsmexlem3  8597
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