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Theorem hsmexlem3 8054
Description: Lemma for hsmex 8058. 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 7286 . . . . 5  |-  ( C  e.  On  ->  C  ~<_*  C )
2 xpwdomg 7299 . . . . 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 7292 . . . 4  |-  ( ( A  X.  C )  ~<_*  ( D  X.  C
)  ->  ~P ( A  X.  C )  ~<_  ~P ( D  X.  C
) )
5 harword 7279 . . . 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 7280 . . . . . 6  |-  Rel  ~<_*
98brrelexi 4729 . . . . 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 8053 . . 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 3181 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   U_ciun 3905   class class class wbr 4023    _E cep 4303   Oncon0 4392    X. cxp 4687   dom cdm 4689   ` cfv 5255    ~<_ cdom 6861  OrdIsocoi 7224  harchar 7270    ~<_* cwdom 7271
This theorem is referenced by:  hsmexlem4  8055  hsmexlem5  8056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-1st 6122  df-2nd 6123  df-riota 6304  df-smo 6363  df-recs 6388  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-oi 7225  df-har 7272  df-wdom 7273
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