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Theorem hsmexlem2 8857
Description: Lemma for hsmex 8862. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9000 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 3988 . . . . . 6  |-  ( B  e.  ~P On  ->  B 
C_  On )
21adantr 466 . . . . 5  |-  ( ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  B  C_  On )
32ralimi 2818 . . . 4  |-  ( A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  A. a  e.  A  B  C_  On )
4 iunss 4337 . . . 4  |-  ( U_ a  e.  A  B  C_  On  <->  A. a  e.  A  B  C_  On )
53, 4sylibr 215 . . 3  |-  ( A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
)  ->  U_ a  e.  A  B  C_  On )
653ad2ant3 1028 . 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 6603 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  On )  ->  ( A  X.  C
)  e.  _V )
873adant3 1025 . . 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 1751 . . . . . . . . 9  |-  F/ a  C  e.  On
10 nfra1 2806 . . . . . . . . 9  |-  F/ a A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C )
119, 10nfan 1984 . . . . . . . 8  |-  F/ a ( C  e.  On  /\ 
A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C ) )
12 rsp 2791 . . . . . . . . 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 5480 . . . . . . . . . . . . . 14  |-  ( C  e.  On  ->  ( dom  F  e.  C  ->  dom  F  C_  C )
)
1413imp 430 . . . . . . . . . . . . 13  |-  ( ( C  e.  On  /\  dom  F  e.  C )  ->  dom  F  C_  C
)
1514adantrl 720 . . . . . . . . . . . 12  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  dom  F  C_  C )
16153adant3 1025 . . . . . . . . . . 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 8057 . . . . . . . . . . . . . . . . . 18  |-  ( B 
C_  On  ->  ( Smo 
F  /\  ran  F  =  B ) )
191, 18syl 17 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  ~P On  ->  ( Smo  F  /\  ran  F  =  B ) )
2019ad2antrl 732 . . . . . . . . . . . . . . . 16  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  -> 
( Smo  F  /\  ran  F  =  B ) )
2120simprd 464 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  ran  F  =  B )
2217oif 8047 . . . . . . . . . . . . . . 15  |-  F : dom  F --> B
2321, 22jctil 539 . . . . . . . . . . . . . 14  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  -> 
( F : dom  F --> B  /\  ran  F  =  B ) )
24 dffo2 5810 . . . . . . . . . . . . . 14  |-  ( F : dom  F -onto-> B  <->  ( F : dom  F --> B  /\  ran  F  =  B ) )
2523, 24sylibr 215 . . . . . . . . . . . . 13  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  F : dom  F -onto-> B
)
26 dffo3 6048 . . . . . . . . . . . . . 14  |-  ( F : dom  F -onto-> B  <->  ( F : dom  F --> B  /\  A. b  e.  B  E. e  e. 
dom  F  b  =  ( F `  e ) ) )
2726simprbi 465 . . . . . . . . . . . . 13  |-  ( F : dom  F -onto-> B  ->  A. b  e.  B  E. e  e.  dom  F  b  =  ( F `
 e ) )
28 rsp 2791 . . . . . . . . . . . . 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 1202 . . . . . . . . . . 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 3526 . . . . . . . . . . 11  |-  ( dom 
F  C_  C  ->  ( E. e  e.  dom  F  b  =  ( F `
 e )  ->  E. e  e.  C  b  =  ( F `  e ) ) )
3216, 30, 31sylc 62 . . . . . . . . . 10  |-  ( ( C  e.  On  /\  ( B  e.  ~P On  /\  dom  F  e.  C )  /\  b  e.  B )  ->  E. e  e.  C  b  =  ( F `  e ) )
33323exp 1204 . . . . . . . . 9  |-  ( C  e.  On  ->  (
( B  e.  ~P On  /\  dom  F  e.  C )  ->  (
b  e.  B  ->  E. e  e.  C  b  =  ( F `  e ) ) ) )
3412, 33sylan9r 662 . . . . . . . 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 2894 . . . . . . 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 1023 . . . . . 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 1751 . . . . . . 7  |-  F/ d E. e  e.  C  b  =  ( F `  e )
38 nfcv 2584 . . . . . . . 8  |-  F/_ a C
39 nfcv 2584 . . . . . . . . . . 11  |-  F/_ a  _E
40 nfcsb1v 3411 . . . . . . . . . . 11  |-  F/_ a [_ d  /  a ]_ B
4139, 40nfoi 8031 . . . . . . . . . 10  |-  F/_ aOrdIso (  _E  ,  [_ d  /  a ]_ B
)
42 nfcv 2584 . . . . . . . . . 10  |-  F/_ a
e
4341, 42nffv 5884 . . . . . . . . 9  |-  F/_ a
(OrdIso (  _E  ,  [_ d  /  a ]_ B ) `  e
)
4443nfeq2 2601 . . . . . . . 8  |-  F/ a  b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e )
4538, 44nfrex 2888 . . . . . . 7  |-  F/ a E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e )
46 csbeq1a 3404 . . . . . . . . . . . 12  |-  ( a  =  d  ->  B  =  [_ d  /  a ]_ B )
47 oieq2 8030 . . . . . . . . . . . 12  |-  ( B  =  [_ d  / 
a ]_ B  -> OrdIso (  _E  ,  B )  = OrdIso
(  _E  ,  [_ d  /  a ]_ B
) )
4846, 47syl 17 . . . . . . . . . . 11  |-  ( a  =  d  -> OrdIso (  _E  ,  B )  = OrdIso
(  _E  ,  [_ d  /  a ]_ B
) )
4917, 48syl5eq 2475 . . . . . . . . . 10  |-  ( a  =  d  ->  F  = OrdIso (  _E  ,  [_ d  /  a ]_ B
) )
5049fveq1d 5879 . . . . . . . . 9  |-  ( a  =  d  ->  ( F `  e )  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B ) `  e
) )
5150eqeq2d 2436 . . . . . . . 8  |-  ( a  =  d  ->  (
b  =  ( F `
 e )  <->  b  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B
) `  e )
) )
5251rexbidv 2939 . . . . . . 7  |-  ( a  =  d  ->  ( E. e  e.  C  b  =  ( F `  e )  <->  E. e  e.  C  b  =  (OrdIso (  _E  ,  [_ d  /  a ]_ B
) `  e )
) )
5337, 45, 52cbvrex 3052 . . . . . 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 229 . . . . 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 4301 . . . . 5  |-  ( b  e.  U_ a  e.  A  B  <->  E. a  e.  A  b  e.  B )
56 vex 3084 . . . . . . . . . . 11  |-  d  e. 
_V
57 vex 3084 . . . . . . . . . . 11  |-  e  e. 
_V
5856, 57op1std 6813 . . . . . . . . . 10  |-  ( c  =  <. d ,  e
>.  ->  ( 1st `  c
)  =  d )
5958csbeq1d 3402 . . . . . . . . 9  |-  ( c  =  <. d ,  e
>.  ->  [_ ( 1st `  c
)  /  a ]_ B  =  [_ d  / 
a ]_ B )
60 oieq2 8030 . . . . . . . . 9  |-  ( [_ ( 1st `  c )  /  a ]_ B  =  [_ d  /  a ]_ B  -> OrdIso (  _E  ,  [_ ( 1st `  c )  /  a ]_ B )  = OrdIso (  _E  ,  [_ d  / 
a ]_ B ) )
6159, 60syl 17 . . . . . . . 8  |-  ( c  =  <. d ,  e
>.  -> OrdIso (  _E  ,  [_ ( 1st `  c )  /  a ]_ B
)  = OrdIso (  _E  ,  [_ d  /  a ]_ B ) )
6256, 57op2ndd 6814 . . . . . . . 8  |-  ( c  =  <. d ,  e
>.  ->  ( 2nd `  c
)  =  e )
6361, 62fveq12d 5883 . . . . . . 7  |-  ( c  =  <. d ,  e
>.  ->  (OrdIso (  _E  ,  [_ ( 1st `  c
)  /  a ]_ B ) `  ( 2nd `  c ) )  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e ) )
6463eqeq2d 2436 . . . . . 6  |-  ( c  =  <. d ,  e
>.  ->  ( b  =  (OrdIso (  _E  ,  [_ ( 1st `  c
)  /  a ]_ B ) `  ( 2nd `  c ) )  <-> 
b  =  (OrdIso (  _E  ,  [_ d  / 
a ]_ B ) `  e ) ) )
6564rexxp 4992 . . . . 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 273 . . . 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 430 . . 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 8098 . 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 8856 . 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 665 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 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776   _Vcvv 3081   [_csb 3395    C_ wss 3436   ~Pcpw 3979   <.cop 4002   U_ciun 4296   class class class wbr 4420    _E cep 4758    X. cxp 4847   dom cdm 4849   ran crn 4850   Oncon0 5438   -->wf 5593   -onto->wfo 5595   ` cfv 5597   1stc1st 6801   2ndc2nd 6802   Smo wsmo 7068  OrdIsocoi 8026  harchar 8073    ~<_* cwdom 8074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-smo 7069  df-recs 7094  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-oi 8027  df-har 8075  df-wdom 8076
This theorem is referenced by:  hsmexlem3  8858
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