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Theorem brwdom 7894
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 3087 . 2  |-  ( Y  e.  V  ->  Y  e.  _V )
2 relwdom 7893 . . . . 5  |-  Rel  ~<_*
32brrelexi 4988 . . . 4  |-  ( X  ~<_*  Y  ->  X  e.  _V )
43a1i 11 . . 3  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  ->  X  e.  _V ) )
5 0ex 4531 . . . . . 6  |-  (/)  e.  _V
6 eleq1a 2537 . . . . . 6  |-  ( (/)  e.  _V  ->  ( X  =  (/)  ->  X  e.  _V ) )
75, 6ax-mp 5 . . . . 5  |-  ( X  =  (/)  ->  X  e. 
_V )
8 forn 5732 . . . . . . 7  |-  ( z : Y -onto-> X  ->  ran  z  =  X
)
9 vex 3081 . . . . . . . 8  |-  z  e. 
_V
109rnex 6623 . . . . . . 7  |-  ran  z  e.  _V
118, 10syl6eqelr 2551 . . . . . 6  |-  ( z : Y -onto-> X  ->  X  e.  _V )
1211exlimiv 1689 . . . . 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 2458 . . . . . 6  |-  ( x  =  X  ->  (
x  =  (/)  <->  X  =  (/) ) )
16 foeq3 5727 . . . . . . 7  |-  ( x  =  X  ->  (
z : y -onto-> x  <-> 
z : y -onto-> X ) )
1716exbidv 1681 . . . . . 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 5726 . . . . . . 7  |-  ( y  =  Y  ->  (
z : y -onto-> X  <-> 
z : Y -onto-> X
) )
2019exbidv 1681 . . . . . 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 7886 . . . . 5  |-  ~<_*  =  { <. x ,  y >.  |  ( x  =  (/)  \/  E. z  z : y
-onto-> x ) }
2318, 21, 22brabg 4717 . . . 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 1370   E.wex 1587    e. wcel 1758   _Vcvv 3078   (/)c0 3746   class class class wbr 4401   ran crn 4950   -onto->wfo 5525    ~<_* cwdom 7884
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 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
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 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-cnv 4957  df-dm 4959  df-rn 4960  df-fn 5530  df-fo 5533  df-wdom 7886
This theorem is referenced by:  brwdomi  7895  brwdomn0  7896  0wdom  7897  fowdom  7898  domwdom  7901  wdomnumr  8346
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