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Theorem hsmexlem3 8300
Description: Lemma for hsmex 8304. 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 7532 . . . . 5  |-  ( C  e.  On  ->  C  ~<_*  C )
2 xpwdomg 7545 . . . . 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 7538 . . . 4  |-  ( ( A  X.  C )  ~<_*  ( D  X.  C
)  ->  ~P ( A  X.  C )  ~<_  ~P ( D  X.  C
) )
5 harword 7525 . . . 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 7526 . . . . . 6  |-  Rel  ~<_*
98brrelexi 4910 . . . . 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 8299 . . 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 3341 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 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   U_ciun 4085   class class class wbr 4204    _E cep 4484   Oncon0 4573    X. cxp 4868   dom cdm 4870   ` cfv 5446    ~<_ cdom 7099  OrdIsocoi 7470  harchar 7516    ~<_* cwdom 7517
This theorem is referenced by:  hsmexlem4  8301  hsmexlem5  8302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-1st 6341  df-2nd 6342  df-riota 6541  df-smo 6600  df-recs 6625  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-oi 7471  df-har 7518  df-wdom 7519
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