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Theorem brwdomn0 7899
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 7896 . . . 4  |-  Rel  ~<_*
21brrelex2i 4991 . . 3  |-  ( X  ~<_*  Y  ->  Y  e.  _V )
32a1i 11 . 2  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  ->  Y  e.  _V ) )
4 fof 5731 . . . . . 6  |-  ( z : Y -onto-> X  -> 
z : Y --> X )
5 fdm 5674 . . . . . 6  |-  ( z : Y --> X  ->  dom  z  =  Y
)
64, 5syl 16 . . . . 5  |-  ( z : Y -onto-> X  ->  dom  z  =  Y
)
7 vex 3081 . . . . . 6  |-  z  e. 
_V
87dmex 6624 . . . . 5  |-  dom  z  e.  _V
96, 8syl6eqelr 2551 . . . 4  |-  ( z : Y -onto-> X  ->  Y  e.  _V )
109exlimiv 1689 . . 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 7897 . . . 4  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
13 df-ne 2650 . . . . . 6  |-  ( X  =/=  (/)  <->  -.  X  =  (/) )
14 biorf 405 . . . . . 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 700 . . 3  |-  ( ( X  =/=  (/)  /\  Y  e.  _V )  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
1817ex 434 . 2  |-  ( X  =/=  (/)  ->  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) ) )
193, 11, 18pm5.21ndd 354 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 368    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2648   _Vcvv 3078   (/)c0 3748   class class class wbr 4403   dom cdm 4951   -->wf 5525   -onto->wfo 5527    ~<_* cwdom 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-dm 4961  df-rn 4962  df-fn 5532  df-f 5533  df-fo 5535  df-wdom 7889
This theorem is referenced by:  brwdom2  7903  wdomtr  7905  wdompwdom  7908  canthwdom  7909  wdomfil  8346  fin1a2lem7  8690
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