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Theorem hsmexlem2 8807
Description: Lemma for hsmex 8812. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 8950 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 4019 . . . . . 6  |-  ( B  e.  ~P On  ->  B 
C_  On )
21adantr 465 . . . . 5  |-  ( ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  B  C_  On )
32ralimi 2857 . . . 4  |-  ( A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  A. a  e.  A  B  C_  On )
4 iunss 4366 . . . 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 1019 . 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 6586 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  On )  ->  ( A  X.  C
)  e.  _V )
873adant3 1016 . . 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 1683 . . . . . . . . 9  |-  F/ a  C  e.  On
10 nfra1 2845 . . . . . . . . 9  |-  F/ a A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C )
119, 10nfan 1875 . . . . . . . 8  |-  F/ a ( C  e.  On  /\ 
A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C ) )
12 rsp 2830 . . . . . . . . 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 4920 . . . . . . . . . . . . . 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 1016 . . . . . . . . . . 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 7965 . . . . . . . . . . . . . . . . . 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 7955 . . . . . . . . . . . . . . 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 5799 . . . . . . . . . . . . . 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 6036 . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . 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 1193 . . . . . . . . . . 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 3565 . . . . . . . . . . 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 1195 . . . . . . . . 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 2933 . . . . . . 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 1014 . . . . . 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 1683 . . . . . . 7  |-  F/ d E. e  e.  C  b  =  ( F `  e )
38 nfcv 2629 . . . . . . . 8  |-  F/_ a C
39 nfcv 2629 . . . . . . . . . . 11  |-  F/_ a  _E
40 nfcsb1v 3451 . . . . . . . . . . 11  |-  F/_ a [_ d  /  a ]_ B
4139, 40nfoi 7939 . . . . . . . . . 10  |-  F/_ aOrdIso (  _E  ,  [_ d  /  a ]_ B
)
42 nfcv 2629 . . . . . . . . . 10  |-  F/_ a
e
4341, 42nffv 5873 . . . . . . . . 9  |-  F/_ a
(OrdIso (  _E  ,  [_ d  /  a ]_ B ) `  e
)
4443nfeq2 2646 . . . . . . . 8  |-  F/ a  b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e )
4538, 44nfrex 2927 . . . . . . 7  |-  F/ a E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e )
46 csbeq1a 3444 . . . . . . . . . . . 12  |-  ( a  =  d  ->  B  =  [_ d  /  a ]_ B )
47 oieq2 7938 . . . . . . . . . . . 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 2520 . . . . . . . . . 10  |-  ( a  =  d  ->  F  = OrdIso (  _E  ,  [_ d  /  a ]_ B
) )
5049fveq1d 5868 . . . . . . . . 9  |-  ( a  =  d  ->  ( F `  e )  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B ) `  e
) )
5150eqeq2d 2481 . . . . . . . 8  |-  ( a  =  d  ->  (
b  =  ( F `
 e )  <->  b  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B
) `  e )
) )
5251rexbidv 2973 . . . . . . 7  |-  ( a  =  d  ->  ( E. e  e.  C  b  =  ( F `  e )  <->  E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B
) `  e )
) )
5337, 45, 52cbvrex 3085 . . . . . 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 4330 . . . . 5  |-  ( b  e.  U_ a  e.  A  B  <->  E. a  e.  A  b  e.  B )
56 vex 3116 . . . . . . . . . . 11  |-  d  e. 
_V
57 vex 3116 . . . . . . . . . . 11  |-  e  e. 
_V
5856, 57op1std 6794 . . . . . . . . . 10  |-  ( c  =  <. d ,  e
>.  ->  ( 1st `  c
)  =  d )
5958csbeq1d 3442 . . . . . . . . 9  |-  ( c  =  <. d ,  e
>.  ->  [_ ( 1st `  c
)  /  a ]_ B  =  [_ d  / 
a ]_ B )
60 oieq2 7938 . . . . . . . . 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 6795 . . . . . . . 8  |-  ( c  =  <. d ,  e
>.  ->  ( 2nd `  c
)  =  e )
6361, 62fveq12d 5872 . . . . . . 7  |-  ( c  =  <. d ,  e
>.  ->  (OrdIso (  _E  ,  [_ ( 1st `  c
)  /  a ]_ B ) `  ( 2nd `  c ) )  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e ) )
6463eqeq2d 2481 . . . . . 6  |-  ( c  =  <. d ,  e
>.  ->  ( b  =  (OrdIso (  _E  ,  [_ ( 1st `  c
)  /  a ]_ B ) `  ( 2nd `  c ) )  <-> 
b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e ) ) )
6564rexxp 5145 . . . . 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 8007 . 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 8806 . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   [_csb 3435    C_ wss 3476   ~Pcpw 4010   <.cop 4033   U_ciun 4325   class class class wbr 4447    _E cep 4789   Oncon0 4878    X. cxp 4997   dom cdm 4999   ran crn 5000   -->wf 5584   -onto->wfo 5586   ` cfv 5588   1stc1st 6782   2ndc2nd 6783   Smo wsmo 7016  OrdIsocoi 7934  harchar 7982    ~<_* cwdom 7983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-1st 6784  df-2nd 6785  df-smo 7017  df-recs 7042  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-oi 7935  df-har 7984  df-wdom 7985
This theorem is referenced by:  hsmexlem3  8808
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