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Theorem hsmexlem3 8070
Description: Lemma for hsmex 8074. 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 7302 . . . . 5  |-  ( C  e.  On  ->  C  ~<_*  C )
2 xpwdomg 7315 . . . . 5  |-  ( ( A  ~<_*  D  /\  C  ~<_*  C
)  ->  ( A  X.  C )  ~<_*  ( D  X.  C
) )
31, 2sylan2 460 . . . 4  |-  ( ( A  ~<_*  D  /\  C  e.  On )  ->  ( A  X.  C )  ~<_*  ( D  X.  C ) )
4 wdompwdom 7308 . . . 4  |-  ( ( A  X.  C )  ~<_*  ( D  X.  C
)  ->  ~P ( A  X.  C )  ~<_  ~P ( D  X.  C
) )
5 harword 7295 . . . 4  |-  ( ~P ( A  X.  C
)  ~<_  ~P ( D  X.  C )  ->  (har `  ~P ( A  X.  C ) )  C_  (har `  ~P ( D  X.  C ) ) )
63, 4, 53syl 18 . . 3  |-  ( ( A  ~<_*  D  /\  C  e.  On )  ->  (har `  ~P ( A  X.  C ) )  C_  (har `  ~P ( D  X.  C ) ) )
76adantr 451 . 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 7296 . . . . . 6  |-  Rel  ~<_*
98brrelexi 4745 . . . . 5  |-  ( A  ~<_*  D  ->  A  e.  _V )
109adantr 451 . . . 4  |-  ( ( A  ~<_*  D  /\  C  e.  On )  ->  A  e.  _V )
1110adantr 451 . . 3  |-  ( ( ( A  ~<_*  D  /\  C  e.  On )  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  A  e.  _V )
12 simplr 731 . . 3  |-  ( ( ( A  ~<_*  D  /\  C  e.  On )  /\  A. a  e.  A  ( B  e.  ~P On  /\ 
dom  F  e.  C
) )  ->  C  e.  On )
13 simpr 447 . . 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 8069 . . 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 1182 . 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 3194 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 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U_ciun 3921   class class class wbr 4039    _E cep 4319   Oncon0 4408    X. cxp 4703   dom cdm 4705   ` cfv 5271    ~<_ cdom 6877  OrdIsocoi 7240  harchar 7286    ~<_* cwdom 7287
This theorem is referenced by:  hsmexlem4  8071  hsmexlem5  8072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-1st 6138  df-2nd 6139  df-riota 6320  df-smo 6379  df-recs 6404  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-oi 7241  df-har 7288  df-wdom 7289
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