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Theorem brwdomn0 7566
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 7563 . . . 4  |-  Rel  ~<_*
21brrelex2i 4948 . . 3  |-  ( X  ~<_*  Y  ->  Y  e.  _V )
32a1i 11 . 2  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  ->  Y  e.  _V ) )
4 fof 5682 . . . . . 6  |-  ( z : Y -onto-> X  -> 
z : Y --> X )
5 fdm 5624 . . . . . 6  |-  ( z : Y --> X  ->  dom  z  =  Y
)
64, 5syl 16 . . . . 5  |-  ( z : Y -onto-> X  ->  dom  z  =  Y
)
7 vex 2965 . . . . . 6  |-  z  e. 
_V
87dmex 5161 . . . . 5  |-  dom  z  e.  _V
96, 8syl6eqelr 2531 . . . 4  |-  ( z : Y -onto-> X  ->  Y  e.  _V )
109exlimiv 1645 . . 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 7564 . . . 4  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
13 df-ne 2607 . . . . . 6  |-  ( X  =/=  (/)  <->  -.  X  =  (/) )
14 biorf 396 . . . . . 6  |-  ( -.  X  =  (/)  ->  ( E. z  z : Y -onto-> X  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
1513, 14sylbi 189 . . . . 5  |-  ( X  =/=  (/)  ->  ( E. z  z : Y -onto-> X 
<->  ( X  =  (/)  \/ 
E. z  z : Y -onto-> X ) ) )
1615bicomd 194 . . . 4  |-  ( X  =/=  (/)  ->  ( ( X  =  (/)  \/  E. z  z : Y -onto-> X )  <->  E. z 
z : Y -onto-> X
) )
1712, 16sylan9bbr 683 . . 3  |-  ( ( X  =/=  (/)  /\  Y  e.  _V )  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
1817ex 425 . 2  |-  ( X  =/=  (/)  ->  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) ) )
193, 11, 18pm5.21ndd 345 1  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359   E.wex 1551    = wceq 1653    e. wcel 1727    =/= wne 2605   _Vcvv 2962   (/)c0 3613   class class class wbr 4237   dom cdm 4907   -->wf 5479   -onto->wfo 5481    ~<_* cwdom 7554
This theorem is referenced by:  brwdom2  7570  wdomtr  7572  wdompwdom  7575  canthwdom  7576  wdomfil  7973  fin1a2lem7  8317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-xp 4913  df-rel 4914  df-cnv 4915  df-dm 4917  df-rn 4918  df-fn 5486  df-f 5487  df-fo 5489  df-wdom 7556
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