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Theorem brwdom 7982
Description: Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.)
Assertion
Ref Expression
brwdom  |-  ( Y  e.  V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
Distinct variable groups:    z, X    z, Y
Allowed substitution hint:    V( z)

Proof of Theorem brwdom
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3115 . 2  |-  ( Y  e.  V  ->  Y  e.  _V )
2 relwdom 7981 . . . . 5  |-  Rel  ~<_*
32brrelexi 5032 . . . 4  |-  ( X  ~<_*  Y  ->  X  e.  _V )
43a1i 11 . . 3  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  ->  X  e.  _V ) )
5 0ex 4570 . . . . . 6  |-  (/)  e.  _V
6 eleq1a 2543 . . . . . 6  |-  ( (/)  e.  _V  ->  ( X  =  (/)  ->  X  e.  _V ) )
75, 6ax-mp 5 . . . . 5  |-  ( X  =  (/)  ->  X  e. 
_V )
8 forn 5789 . . . . . . 7  |-  ( z : Y -onto-> X  ->  ran  z  =  X
)
9 vex 3109 . . . . . . . 8  |-  z  e. 
_V
109rnex 6708 . . . . . . 7  |-  ran  z  e.  _V
118, 10syl6eqelr 2557 . . . . . 6  |-  ( z : Y -onto-> X  ->  X  e.  _V )
1211exlimiv 1693 . . . . 5  |-  ( E. z  z : Y -onto-> X  ->  X  e.  _V )
137, 12jaoi 379 . . . 4  |-  ( ( X  =  (/)  \/  E. z  z : Y -onto-> X )  ->  X  e.  _V )
1413a1i 11 . . 3  |-  ( Y  e.  _V  ->  (
( X  =  (/)  \/ 
E. z  z : Y -onto-> X )  ->  X  e.  _V ) )
15 eqeq1 2464 . . . . . 6  |-  ( x  =  X  ->  (
x  =  (/)  <->  X  =  (/) ) )
16 foeq3 5784 . . . . . . 7  |-  ( x  =  X  ->  (
z : y -onto-> x  <-> 
z : y -onto-> X ) )
1716exbidv 1685 . . . . . 6  |-  ( x  =  X  ->  ( E. z  z :
y -onto-> x  <->  E. z  z : y -onto-> X ) )
1815, 17orbi12d 709 . . . . 5  |-  ( x  =  X  ->  (
( x  =  (/)  \/ 
E. z  z : y -onto-> x )  <->  ( X  =  (/)  \/  E. z 
z : y -onto-> X ) ) )
19 foeq2 5783 . . . . . . 7  |-  ( y  =  Y  ->  (
z : y -onto-> X  <-> 
z : Y -onto-> X
) )
2019exbidv 1685 . . . . . 6  |-  ( y  =  Y  ->  ( E. z  z :
y -onto-> X  <->  E. z  z : Y -onto-> X ) )
2120orbi2d 701 . . . . 5  |-  ( y  =  Y  ->  (
( X  =  (/)  \/ 
E. z  z : y -onto-> X )  <->  ( X  =  (/)  \/  E. z 
z : Y -onto-> X
) ) )
22 df-wdom 7974 . . . . 5  |-  ~<_*  =  { <. x ,  y >.  |  ( x  =  (/)  \/  E. z  z : y
-onto-> x ) }
2318, 21, 22brabg 4759 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
2423expcom 435 . . 3  |-  ( Y  e.  _V  ->  ( X  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) ) )
254, 14, 24pm5.21ndd 354 . 2  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
261, 25syl 16 1  |-  ( Y  e.  V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1374   E.wex 1591    e. wcel 1762   _Vcvv 3106   (/)c0 3778   class class class wbr 4440   ran crn 4993   -onto->wfo 5577    ~<_* cwdom 7972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-dm 5002  df-rn 5003  df-fn 5582  df-fo 5585  df-wdom 7974
This theorem is referenced by:  brwdomi  7983  brwdomn0  7984  0wdom  7985  fowdom  7986  domwdom  7989  wdomnumr  8434
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