MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wdompwdom Structured version   Unicode version

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 5041 . . . . 5  |-  ( X  ~<_*  Y  ->  Y  e.  _V )
3 pwexg 4631 . . . . 5  |-  ( Y  e.  _V  ->  ~P Y  e.  _V )
42, 3syl 16 . . . 4  |-  ( X  ~<_*  Y  ->  ~P Y  e. 
_V )
5 0ss 3814 . . . . 5  |-  (/)  C_  Y
6 sspwb 4696 . . . . 5  |-  ( (/)  C_  Y  <->  ~P (/)  C_  ~P Y
)
75, 6mpbi 208 . . . 4  |-  ~P (/)  C_  ~P Y
8 ssdomg 7561 . . . 4  |-  ( ~P Y  e.  _V  ->  ( ~P (/)  C_  ~P Y  ->  ~P (/)  ~<_  ~P Y
) )
94, 7, 8mpisyl 18 . . 3  |-  ( X  ~<_*  Y  ->  ~P (/)  ~<_  ~P Y
)
10 pweq 4013 . . . 4  |-  ( X  =  (/)  ->  ~P X  =  ~P (/) )
1110breq1d 4457 . . 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 3116 . . . . 5  |-  z  e. 
_V
15 fopwdom 7625 . . . . 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 1698 . . 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 2780 1  |-  ( X  ~<_*  Y  ->  ~P X  ~<_  ~P Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447   -onto->wfo 5586    ~<_ cdom 7514    ~<_* cwdom 7983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-dom 7518  df-wdom 7985
This theorem is referenced by:  isfin32i  8745  hsmexlem1  8806  hsmexlem3  8808  gchhar  9057
  Copyright terms: Public domain W3C validator