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Theorem hsmexlem3 8602
Description: Lemma for hsmex 8606. Clear  I hypothesis and extend previous result by dominance. Note that this could be substantially strengthened, e.g. using the weak Hartogs function, but all we need here is that there be *some* dominating ordinal. (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
hsmexlem3  |-  ( ( ( A  ~<_*  D  /\  C  e.  On )  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  dom  G  e.  (har `  ~P ( D  X.  C
) ) )
Distinct variable groups:    A, a    C, a
Allowed substitution hints:    B( a)    D( a)    F( a)    G( a)

Proof of Theorem hsmexlem3
StepHypRef Expression
1 wdomref 7792 . . . . 5  |-  ( C  e.  On  ->  C  ~<_*  C )
2 xpwdomg 7805 . . . . 5  |-  ( ( A  ~<_*  D  /\  C  ~<_*  C
)  ->  ( A  X.  C )  ~<_*  ( D  X.  C
) )
31, 2sylan2 474 . . . 4  |-  ( ( A  ~<_*  D  /\  C  e.  On )  ->  ( A  X.  C )  ~<_*  ( D  X.  C ) )
4 wdompwdom 7798 . . . 4  |-  ( ( A  X.  C )  ~<_*  ( D  X.  C
)  ->  ~P ( A  X.  C )  ~<_  ~P ( D  X.  C
) )
5 harword 7785 . . . 4  |-  ( ~P ( A  X.  C
)  ~<_  ~P ( D  X.  C )  ->  (har `  ~P ( A  X.  C ) )  C_  (har `  ~P ( D  X.  C ) ) )
63, 4, 53syl 20 . . 3  |-  ( ( A  ~<_*  D  /\  C  e.  On )  ->  (har `  ~P ( A  X.  C ) )  C_  (har `  ~P ( D  X.  C ) ) )
76adantr 465 . 2  |-  ( ( ( A  ~<_*  D  /\  C  e.  On )  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  (har `  ~P ( A  X.  C ) )  C_  (har `  ~P ( D  X.  C ) ) )
8 relwdom 7786 . . . . . 6  |-  Rel  ~<_*
98brrelexi 4884 . . . . 5  |-  ( A  ~<_*  D  ->  A  e.  _V )
109adantr 465 . . . 4  |-  ( ( A  ~<_*  D  /\  C  e.  On )  ->  A  e.  _V )
1110adantr 465 . . 3  |-  ( ( ( A  ~<_*  D  /\  C  e.  On )  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  A  e.  _V )
12 simplr 754 . . 3  |-  ( ( ( A  ~<_*  D  /\  C  e.  On )  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  C  e.  On )
13 simpr 461 . . 3  |-  ( ( ( A  ~<_*  D  /\  C  e.  On )  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C ) )
14 hsmexlem.f . . . 4  |-  F  = OrdIso
(  _E  ,  B
)
15 hsmexlem.g . . . 4  |-  G  = OrdIso
(  _E  ,  U_ a  e.  A  B
)
1614, 15hsmexlem2 8601 . . 3  |-  ( ( 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
) ) )
1711, 12, 13, 16syl3anc 1218 . 2  |-  ( ( ( A  ~<_*  D  /\  C  e.  On )  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  dom  G  e.  (har `  ~P ( A  X.  C
) ) )
187, 17sseldd 3362 1  |-  ( ( ( A  ~<_*  D  /\  C  e.  On )  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  dom  G  e.  (har `  ~P ( D  X.  C
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   _Vcvv 2977    C_ wss 3333   ~Pcpw 3865   U_ciun 4176   class class class wbr 4297    _E cep 4635   Oncon0 4724    X. cxp 4843   dom cdm 4845   ` cfv 5423    ~<_ cdom 7313  OrdIsocoi 7728  harchar 7776    ~<_* cwdom 7777
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-1st 6582  df-2nd 6583  df-smo 6812  df-recs 6837  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-oi 7729  df-har 7778  df-wdom 7779
This theorem is referenced by:  hsmexlem4  8603  hsmexlem5  8604
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