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Theorem brwdom 7297
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 2809 . 2  |-  ( Y  e.  V  ->  Y  e.  _V )
2 relwdom 7296 . . . . 5  |-  Rel  ~<_*
32brrelexi 4745 . . . 4  |-  ( X  ~<_*  Y  ->  X  e.  _V )
43a1i 10 . . 3  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  ->  X  e.  _V ) )
5 0ex 4166 . . . . . 6  |-  (/)  e.  _V
6 eleq1a 2365 . . . . . 6  |-  ( (/)  e.  _V  ->  ( X  =  (/)  ->  X  e.  _V ) )
75, 6ax-mp 8 . . . . 5  |-  ( X  =  (/)  ->  X  e. 
_V )
8 forn 5470 . . . . . . 7  |-  ( z : Y -onto-> X  ->  ran  z  =  X
)
9 vex 2804 . . . . . . . 8  |-  z  e. 
_V
109rnex 4958 . . . . . . 7  |-  ran  z  e.  _V
118, 10syl6eqelr 2385 . . . . . 6  |-  ( z : Y -onto-> X  ->  X  e.  _V )
1211exlimiv 1624 . . . . 5  |-  ( E. z  z : Y -onto-> X  ->  X  e.  _V )
137, 12jaoi 368 . . . 4  |-  ( ( X  =  (/)  \/  E. z  z : Y -onto-> X )  ->  X  e.  _V )
1413a1i 10 . . 3  |-  ( Y  e.  _V  ->  (
( X  =  (/)  \/ 
E. z  z : Y -onto-> X )  ->  X  e.  _V ) )
15 eqeq1 2302 . . . . . 6  |-  ( x  =  X  ->  (
x  =  (/)  <->  X  =  (/) ) )
16 foeq3 5465 . . . . . . 7  |-  ( x  =  X  ->  (
z : y -onto-> x  <-> 
z : y -onto-> X ) )
1716exbidv 1616 . . . . . 6  |-  ( x  =  X  ->  ( E. z  z :
y -onto-> x  <->  E. z  z : y -onto-> X ) )
1815, 17orbi12d 690 . . . . 5  |-  ( x  =  X  ->  (
( x  =  (/)  \/ 
E. z  z : y -onto-> x )  <->  ( X  =  (/)  \/  E. z 
z : y -onto-> X ) ) )
19 foeq2 5464 . . . . . . 7  |-  ( y  =  Y  ->  (
z : y -onto-> X  <-> 
z : Y -onto-> X
) )
2019exbidv 1616 . . . . . 6  |-  ( y  =  Y  ->  ( E. z  z :
y -onto-> X  <->  E. z  z : Y -onto-> X ) )
2120orbi2d 682 . . . . 5  |-  ( y  =  Y  ->  (
( X  =  (/)  \/ 
E. z  z : y -onto-> X )  <->  ( X  =  (/)  \/  E. z 
z : Y -onto-> X
) ) )
22 df-wdom 7289 . . . . 5  |-  ~<_*  =  { <. x ,  y >.  |  ( x  =  (/)  \/  E. z  z : y
-onto-> x ) }
2318, 21, 22brabg 4300 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
2423expcom 424 . . 3  |-  ( Y  e.  _V  ->  ( X  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) ) )
254, 14, 24pm5.21ndd 343 . 2  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
261, 25syl 15 1  |-  ( Y  e.  V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   class class class wbr 4039   ran crn 4706   -onto->wfo 5269    ~<_* cwdom 7287
This theorem is referenced by:  brwdomi  7298  brwdomn0  7299  0wdom  7300  fowdom  7301  domwdom  7304  wdomnumr  7707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-fn 5274  df-fo 5277  df-wdom 7289
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