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Theorem brwdomn0 7328
Description: Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
brwdomn0  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
Distinct variable groups:    z, X    z, Y

Proof of Theorem brwdomn0
StepHypRef Expression
1 relwdom 7325 . . . 4  |-  Rel  ~<_*
21brrelex2i 4767 . . 3  |-  ( X  ~<_*  Y  ->  Y  e.  _V )
32a1i 10 . 2  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  ->  Y  e.  _V ) )
4 fof 5489 . . . . . 6  |-  ( z : Y -onto-> X  -> 
z : Y --> X )
5 fdm 5431 . . . . . 6  |-  ( z : Y --> X  ->  dom  z  =  Y
)
64, 5syl 15 . . . . 5  |-  ( z : Y -onto-> X  ->  dom  z  =  Y
)
7 vex 2825 . . . . . 6  |-  z  e. 
_V
87dmex 4978 . . . . 5  |-  dom  z  e.  _V
96, 8syl6eqelr 2405 . . . 4  |-  ( z : Y -onto-> X  ->  Y  e.  _V )
109exlimiv 1625 . . 3  |-  ( E. z  z : Y -onto-> X  ->  Y  e.  _V )
1110a1i 10 . 2  |-  ( X  =/=  (/)  ->  ( E. z  z : Y -onto-> X  ->  Y  e.  _V ) )
12 brwdom 7326 . . . 4  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
13 df-ne 2481 . . . . . 6  |-  ( X  =/=  (/)  <->  -.  X  =  (/) )
14 biorf 394 . . . . . 6  |-  ( -.  X  =  (/)  ->  ( E. z  z : Y -onto-> X  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
1513, 14sylbi 187 . . . . 5  |-  ( X  =/=  (/)  ->  ( E. z  z : Y -onto-> X 
<->  ( X  =  (/)  \/ 
E. z  z : Y -onto-> X ) ) )
1615bicomd 192 . . . 4  |-  ( X  =/=  (/)  ->  ( ( X  =  (/)  \/  E. z  z : Y -onto-> X )  <->  E. z 
z : Y -onto-> X
) )
1712, 16sylan9bbr 681 . . 3  |-  ( ( X  =/=  (/)  /\  Y  e.  _V )  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
1817ex 423 . 2  |-  ( X  =/=  (/)  ->  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) ) )
193, 11, 18pm5.21ndd 343 1  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357   E.wex 1532    = wceq 1633    e. wcel 1701    =/= wne 2479   _Vcvv 2822   (/)c0 3489   class class class wbr 4060   dom cdm 4726   -->wf 5288   -onto->wfo 5290    ~<_* cwdom 7316
This theorem is referenced by:  brwdom2  7332  wdomtr  7334  wdompwdom  7337  canthwdom  7338  wdomfil  7733  fin1a2lem7  8077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733  df-cnv 4734  df-dm 4736  df-rn 4737  df-fn 5295  df-f 5296  df-fo 5298  df-wdom 7318
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