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Theorem brwdomn0 7987
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 7984 . . . 4  |-  Rel  ~<_*
21brrelex2i 5030 . . 3  |-  ( X  ~<_*  Y  ->  Y  e.  _V )
32a1i 11 . 2  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  ->  Y  e.  _V ) )
4 fof 5777 . . . . . 6  |-  ( z : Y -onto-> X  -> 
z : Y --> X )
5 fdm 5717 . . . . . 6  |-  ( z : Y --> X  ->  dom  z  =  Y
)
64, 5syl 16 . . . . 5  |-  ( z : Y -onto-> X  ->  dom  z  =  Y
)
7 vex 3109 . . . . . 6  |-  z  e. 
_V
87dmex 6706 . . . . 5  |-  dom  z  e.  _V
96, 8syl6eqelr 2551 . . . 4  |-  ( z : Y -onto-> X  ->  Y  e.  _V )
109exlimiv 1727 . . 3  |-  ( E. z  z : Y -onto-> X  ->  Y  e.  _V )
1110a1i 11 . 2  |-  ( X  =/=  (/)  ->  ( E. z  z : Y -onto-> X  ->  Y  e.  _V ) )
12 brwdom 7985 . . . 4  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
13 df-ne 2651 . . . . . 6  |-  ( X  =/=  (/)  <->  -.  X  =  (/) )
14 biorf 403 . . . . . 6  |-  ( -.  X  =  (/)  ->  ( E. z  z : Y -onto-> X  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
1513, 14sylbi 195 . . . . 5  |-  ( X  =/=  (/)  ->  ( E. z  z : Y -onto-> X 
<->  ( X  =  (/)  \/ 
E. z  z : Y -onto-> X ) ) )
1615bicomd 201 . . . 4  |-  ( X  =/=  (/)  ->  ( ( X  =  (/)  \/  E. z  z : Y -onto-> X )  <->  E. z 
z : Y -onto-> X
) )
1712, 16sylan9bbr 698 . . 3  |-  ( ( X  =/=  (/)  /\  Y  e.  _V )  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
1817ex 432 . 2  |-  ( X  =/=  (/)  ->  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) ) )
193, 11, 18pm5.21ndd 352 1  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   _Vcvv 3106   (/)c0 3783   class class class wbr 4439   dom cdm 4988   -->wf 5566   -onto->wfo 5568    ~<_* cwdom 7975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-fn 5573  df-f 5574  df-fo 5576  df-wdom 7977
This theorem is referenced by:  brwdom2  7991  wdomtr  7993  wdompwdom  7996  canthwdom  7997  wdomfil  8433  fin1a2lem7  8777
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