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Theorem brwdomi 7298
Description: Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
brwdomi  |-  ( X  ~<_*  Y  ->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) )
Distinct variable groups:    z, X    z, Y

Proof of Theorem brwdomi
StepHypRef Expression
1 relwdom 7296 . . . 4  |-  Rel  ~<_*
21brrelex2i 4746 . . 3  |-  ( X  ~<_*  Y  ->  Y  e.  _V )
3 brwdom 7297 . . 3  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
42, 3syl 15 . 2  |-  ( X  ~<_*  Y  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
54ibi 232 1  |-  ( X  ~<_*  Y  ->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   class class class wbr 4039   -onto->wfo 5269    ~<_* cwdom 7287
This theorem is referenced by:  numwdom  7702
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-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-fn 5274  df-fo 5277  df-wdom 7289
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