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Theorem hsmexlem3 8264
Description: Lemma for hsmex 8268. 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 7496 . . . . 5  |-  ( C  e.  On  ->  C  ~<_*  C )
2 xpwdomg 7509 . . . . 5  |-  ( ( A  ~<_*  D  /\  C  ~<_*  C
)  ->  ( A  X.  C )  ~<_*  ( D  X.  C
) )
31, 2sylan2 461 . . . 4  |-  ( ( A  ~<_*  D  /\  C  e.  On )  ->  ( A  X.  C )  ~<_*  ( D  X.  C ) )
4 wdompwdom 7502 . . . 4  |-  ( ( A  X.  C )  ~<_*  ( D  X.  C
)  ->  ~P ( A  X.  C )  ~<_  ~P ( D  X.  C
) )
5 harword 7489 . . . 4  |-  ( ~P ( A  X.  C
)  ~<_  ~P ( D  X.  C )  ->  (har `  ~P ( A  X.  C ) )  C_  (har `  ~P ( D  X.  C ) ) )
63, 4, 53syl 19 . . 3  |-  ( ( A  ~<_*  D  /\  C  e.  On )  ->  (har `  ~P ( A  X.  C ) )  C_  (har `  ~P ( D  X.  C ) ) )
76adantr 452 . 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 7490 . . . . . 6  |-  Rel  ~<_*
98brrelexi 4877 . . . . 5  |-  ( A  ~<_*  D  ->  A  e.  _V )
109adantr 452 . . . 4  |-  ( ( A  ~<_*  D  /\  C  e.  On )  ->  A  e.  _V )
1110adantr 452 . . 3  |-  ( ( ( A  ~<_*  D  /\  C  e.  On )  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  A  e.  _V )
12 simplr 732 . . 3  |-  ( ( ( A  ~<_*  D  /\  C  e.  On )  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  C  e.  On )
13 simpr 448 . . 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 8263 . . 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 1184 . 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 3309 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   U_ciun 4053   class class class wbr 4172    _E cep 4452   Oncon0 4541    X. cxp 4835   dom cdm 4837   ` cfv 5413    ~<_ cdom 7066  OrdIsocoi 7434  harchar 7480    ~<_* cwdom 7481
This theorem is referenced by:  hsmexlem4  8265  hsmexlem5  8266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-1st 6308  df-2nd 6309  df-riota 6508  df-smo 6567  df-recs 6592  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-oi 7435  df-har 7482  df-wdom 7483
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