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Theorem brwdomn0 7993
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 7990 . . . 4  |-  Rel  ~<_*
21brrelex2i 5027 . . 3  |-  ( X  ~<_*  Y  ->  Y  e.  _V )
32a1i 11 . 2  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  ->  Y  e.  _V ) )
4 fof 5781 . . . . . 6  |-  ( z : Y -onto-> X  -> 
z : Y --> X )
5 fdm 5721 . . . . . 6  |-  ( z : Y --> X  ->  dom  z  =  Y
)
64, 5syl 16 . . . . 5  |-  ( z : Y -onto-> X  ->  dom  z  =  Y
)
7 vex 3096 . . . . . 6  |-  z  e. 
_V
87dmex 6714 . . . . 5  |-  dom  z  e.  _V
96, 8syl6eqelr 2538 . . . 4  |-  ( z : Y -onto-> X  ->  Y  e.  _V )
109exlimiv 1707 . . 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 7991 . . . 4  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
13 df-ne 2638 . . . . . 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 1381   E.wex 1597    e. wcel 1802    =/= wne 2636   _Vcvv 3093   (/)c0 3767   class class class wbr 4433   dom cdm 4985   -->wf 5570   -onto->wfo 5572    ~<_* cwdom 7981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-xp 4991  df-rel 4992  df-cnv 4993  df-dm 4995  df-rn 4996  df-fn 5577  df-f 5578  df-fo 5580  df-wdom 7983
This theorem is referenced by:  brwdom2  7997  wdomtr  7999  wdompwdom  8002  canthwdom  8003  wdomfil  8440  fin1a2lem7  8784
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