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Theorem wdompwdom 8004
Description: Weak dominance strengthens to usual dominance on the power sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
wdompwdom  |-  ( X  ~<_*  Y  ->  ~P X  ~<_  ~P Y )

Proof of Theorem wdompwdom
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 relwdom 7992 . . . . . 6  |-  Rel  ~<_*
21brrelex2i 5028 . . . . 5  |-  ( X  ~<_*  Y  ->  Y  e.  _V )
3 pwexg 4618 . . . . 5  |-  ( Y  e.  _V  ->  ~P Y  e.  _V )
42, 3syl 16 . . . 4  |-  ( X  ~<_*  Y  ->  ~P Y  e. 
_V )
5 0ss 3797 . . . . 5  |-  (/)  C_  Y
6 sspwb 4683 . . . . 5  |-  ( (/)  C_  Y  <->  ~P (/)  C_  ~P Y
)
75, 6mpbi 208 . . . 4  |-  ~P (/)  C_  ~P Y
8 ssdomg 7560 . . . 4  |-  ( ~P Y  e.  _V  ->  ( ~P (/)  C_  ~P Y  ->  ~P (/)  ~<_  ~P Y
) )
94, 7, 8mpisyl 18 . . 3  |-  ( X  ~<_*  Y  ->  ~P (/)  ~<_  ~P Y
)
10 pweq 3997 . . . 4  |-  ( X  =  (/)  ->  ~P X  =  ~P (/) )
1110breq1d 4444 . . 3  |-  ( X  =  (/)  ->  ( ~P X  ~<_  ~P Y  <->  ~P (/)  ~<_  ~P Y
) )
129, 11syl5ibr 221 . 2  |-  ( X  =  (/)  ->  ( X  ~<_*  Y  ->  ~P X  ~<_  ~P Y ) )
13 brwdomn0 7995 . . 3  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
14 vex 3096 . . . . 5  |-  z  e. 
_V
15 fopwdom 7624 . . . . 5  |-  ( ( z  e.  _V  /\  z : Y -onto-> X )  ->  ~P X  ~<_  ~P Y )
1614, 15mpan 670 . . . 4  |-  ( z : Y -onto-> X  ->  ~P X  ~<_  ~P Y
)
1716exlimiv 1707 . . 3  |-  ( E. z  z : Y -onto-> X  ->  ~P X  ~<_  ~P Y )
1813, 17syl6bi 228 . 2  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  ->  ~P X  ~<_  ~P Y ) )
1912, 18pm2.61ine 2754 1  |-  ( X  ~<_*  Y  ->  ~P X  ~<_  ~P Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381   E.wex 1597    e. wcel 1802    =/= wne 2636   _Vcvv 3093    C_ wss 3459   (/)c0 3768   ~Pcpw 3994   class class class wbr 4434   -onto->wfo 5573    ~<_ cdom 7513    ~<_* cwdom 7983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-dom 7517  df-wdom 7985
This theorem is referenced by:  isfin32i  8745  hsmexlem1  8806  hsmexlem3  8808  gchhar  9057
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