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Theorem hsmexlem2 8592
Description: Lemma for hsmex 8597. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 8735 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 3866 . . . . . 6  |-  ( B  e.  ~P On  ->  B 
C_  On )
21adantr 462 . . . . 5  |-  ( ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  B  C_  On )
32ralimi 2789 . . . 4  |-  ( A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  A. a  e.  A  B  C_  On )
4 iunss 4208 . . . 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 1006 . 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 6506 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  On )  ->  ( A  X.  C
)  e.  _V )
873adant3 1003 . . 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 1678 . . . . . . . . 9  |-  F/ a  C  e.  On
10 nfra1 2764 . . . . . . . . 9  |-  F/ a A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C )
119, 10nfan 1865 . . . . . . . 8  |-  F/ a ( C  e.  On  /\ 
A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C ) )
12 rsp 2774 . . . . . . . . 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 4757 . . . . . . . . . . . . . 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 710 . . . . . . . . . . . 12  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  dom  F  C_  C )
16153adant3 1003 . . . . . . . . . . 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 7750 . . . . . . . . . . . . . . . . . 18  |-  ( B 
C_  On  ->  ( Smo 
F  /\  ran  F  =  B ) )
191, 18syl 16 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  ~P On  ->  ( Smo  F  /\  ran  F  =  B ) )
2019ad2antrl 722 . . . . . . . . . . . . . . . 16  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  -> 
( Smo  F  /\  ran  F  =  B ) )
2120simprd 460 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  ran  F  =  B )
2217oif 7740 . . . . . . . . . . . . . . 15  |-  F : dom  F --> B
2321, 22jctil 534 . . . . . . . . . . . . . 14  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  -> 
( F : dom  F --> B  /\  ran  F  =  B ) )
24 dffo2 5621 . . . . . . . . . . . . . 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 5855 . . . . . . . . . . . . . 14  |-  ( F : dom  F -onto-> B  <->  ( F : dom  F --> B  /\  A. b  e.  B  E. e  e. 
dom  F  b  =  ( F `  e ) ) )
2726simprbi 461 . . . . . . . . . . . . 13  |-  ( F : dom  F -onto-> B  ->  A. b  e.  B  E. e  e.  dom  F  b  =  ( F `
 e ) )
28 rsp 2774 . . . . . . . . . . . . 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 1179 . . . . . . . . . . 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 3414 . . . . . . . . . . 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 1181 . . . . . . . . 9  |-  ( C  e.  On  ->  (
( B  e.  ~P On  /\  dom  F  e.  C )  ->  (
b  e.  B  ->  E. e  e.  C  b  =  ( F `  e ) ) ) )
3412, 33sylan9r 653 . . . . . . . 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 2822 . . . . . . 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 1001 . . . . . 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 1678 . . . . . . 7  |-  F/ d E. e  e.  C  b  =  ( F `  e )
38 nfcv 2577 . . . . . . . 8  |-  F/_ a C
39 nfcv 2577 . . . . . . . . . . 11  |-  F/_ a  _E
40 nfcsb1v 3301 . . . . . . . . . . 11  |-  F/_ a [_ d  /  a ]_ B
4139, 40nfoi 7724 . . . . . . . . . 10  |-  F/_ aOrdIso (  _E  ,  [_ d  /  a ]_ B
)
42 nfcv 2577 . . . . . . . . . 10  |-  F/_ a
e
4341, 42nffv 5695 . . . . . . . . 9  |-  F/_ a
(OrdIso (  _E  ,  [_ d  /  a ]_ B ) `  e
)
4443nfeq2 2588 . . . . . . . 8  |-  F/ a  b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e )
4538, 44nfrex 2769 . . . . . . 7  |-  F/ a E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e )
46 csbeq1a 3294 . . . . . . . . . . . 12  |-  ( a  =  d  ->  B  =  [_ d  /  a ]_ B )
47 oieq2 7723 . . . . . . . . . . . 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 2485 . . . . . . . . . 10  |-  ( a  =  d  ->  F  = OrdIso (  _E  ,  [_ d  /  a ]_ B
) )
5049fveq1d 5690 . . . . . . . . 9  |-  ( a  =  d  ->  ( F `  e )  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B ) `  e
) )
5150eqeq2d 2452 . . . . . . . 8  |-  ( a  =  d  ->  (
b  =  ( F `
 e )  <->  b  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B
) `  e )
) )
5251rexbidv 2734 . . . . . . 7  |-  ( a  =  d  ->  ( E. e  e.  C  b  =  ( F `  e )  <->  E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B
) `  e )
) )
5337, 45, 52cbvrex 2942 . . . . . 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 4172 . . . . 5  |-  ( b  e.  U_ a  e.  A  B  <->  E. a  e.  A  b  e.  B )
56 vex 2973 . . . . . . . . . . 11  |-  d  e. 
_V
57 vex 2973 . . . . . . . . . . 11  |-  e  e. 
_V
5856, 57op1std 6586 . . . . . . . . . 10  |-  ( c  =  <. d ,  e
>.  ->  ( 1st `  c
)  =  d )
5958csbeq1d 3292 . . . . . . . . 9  |-  ( c  =  <. d ,  e
>.  ->  [_ ( 1st `  c
)  /  a ]_ B  =  [_ d  / 
a ]_ B )
60 oieq2 7723 . . . . . . . . 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 6587 . . . . . . . 8  |-  ( c  =  <. d ,  e
>.  ->  ( 2nd `  c
)  =  e )
6361, 62fveq12d 5694 . . . . . . 7  |-  ( c  =  <. d ,  e
>.  ->  (OrdIso (  _E  ,  [_ ( 1st `  c
)  /  a ]_ B ) `  ( 2nd `  c ) )  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e ) )
6463eqeq2d 2452 . . . . . 6  |-  ( c  =  <. d ,  e
>.  ->  ( b  =  (OrdIso (  _E  ,  [_ ( 1st `  c
)  /  a ]_ B ) `  ( 2nd `  c ) )  <-> 
b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e ) ) )
6564rexxp 4978 . . . . 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 7792 . 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 8591 . 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 656 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 960    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   _Vcvv 2970   [_csb 3285    C_ wss 3325   ~Pcpw 3857   <.cop 3880   U_ciun 4168   class class class wbr 4289    _E cep 4626   Oncon0 4715    X. cxp 4834   dom cdm 4836   ran crn 4837   -->wf 5411   -onto->wfo 5413   ` cfv 5415   1stc1st 6574   2ndc2nd 6575   Smo wsmo 6802  OrdIsocoi 7719  harchar 7767    ~<_* cwdom 7768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-1st 6576  df-2nd 6577  df-smo 6803  df-recs 6828  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-oi 7720  df-har 7769  df-wdom 7770
This theorem is referenced by:  hsmexlem3  8593
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