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Theorem wdompwdom 7791
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 7779 . . . . . 6  |-  Rel  ~<_*
21brrelex2i 4878 . . . . 5  |-  ( X  ~<_*  Y  ->  Y  e.  _V )
3 pwexg 4474 . . . . 5  |-  ( Y  e.  _V  ->  ~P Y  e.  _V )
42, 3syl 16 . . . 4  |-  ( X  ~<_*  Y  ->  ~P Y  e. 
_V )
5 0ss 3664 . . . . 5  |-  (/)  C_  Y
6 sspwb 4539 . . . . 5  |-  ( (/)  C_  Y  <->  ~P (/)  C_  ~P Y
)
75, 6mpbi 208 . . . 4  |-  ~P (/)  C_  ~P Y
8 ssdomg 7353 . . . 4  |-  ( ~P Y  e.  _V  ->  ( ~P (/)  C_  ~P Y  ->  ~P (/)  ~<_  ~P Y
) )
94, 7, 8mpisyl 18 . . 3  |-  ( X  ~<_*  Y  ->  ~P (/)  ~<_  ~P Y
)
10 pweq 3861 . . . 4  |-  ( X  =  (/)  ->  ~P X  =  ~P (/) )
1110breq1d 4300 . . 3  |-  ( X  =  (/)  ->  ( ~P X  ~<_  ~P Y  <->  ~P (/)  ~<_  ~P Y
) )
129, 11syl5ibr 221 . 2  |-  ( X  =  (/)  ->  ( X  ~<_*  Y  ->  ~P X  ~<_  ~P Y ) )
13 brwdomn0 7782 . . 3  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
14 vex 2973 . . . . 5  |-  z  e. 
_V
15 fopwdom 7417 . . . . 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 1688 . . 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 2685 1  |-  ( X  ~<_*  Y  ->  ~P X  ~<_  ~P Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2604   _Vcvv 2970    C_ wss 3326   (/)c0 3635   ~Pcpw 3858   class class class wbr 4290   -onto->wfo 5414    ~<_ cdom 7306    ~<_* cwdom 7770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-dom 7310  df-wdom 7772
This theorem is referenced by:  isfin32i  8532  hsmexlem1  8593  hsmexlem3  8595  gchhar  8844
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