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Theorem brwdom 7537
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 2966 . 2  |-  ( Y  e.  V  ->  Y  e.  _V )
2 relwdom 7536 . . . . 5  |-  Rel  ~<_*
32brrelexi 4920 . . . 4  |-  ( X  ~<_*  Y  ->  X  e.  _V )
43a1i 11 . . 3  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  ->  X  e.  _V ) )
5 0ex 4341 . . . . . 6  |-  (/)  e.  _V
6 eleq1a 2507 . . . . . 6  |-  ( (/)  e.  _V  ->  ( X  =  (/)  ->  X  e.  _V ) )
75, 6ax-mp 8 . . . . 5  |-  ( X  =  (/)  ->  X  e. 
_V )
8 forn 5658 . . . . . . 7  |-  ( z : Y -onto-> X  ->  ran  z  =  X
)
9 vex 2961 . . . . . . . 8  |-  z  e. 
_V
109rnex 5135 . . . . . . 7  |-  ran  z  e.  _V
118, 10syl6eqelr 2527 . . . . . 6  |-  ( z : Y -onto-> X  ->  X  e.  _V )
1211exlimiv 1645 . . . . 5  |-  ( E. z  z : Y -onto-> X  ->  X  e.  _V )
137, 12jaoi 370 . . . 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 2444 . . . . . 6  |-  ( x  =  X  ->  (
x  =  (/)  <->  X  =  (/) ) )
16 foeq3 5653 . . . . . . 7  |-  ( x  =  X  ->  (
z : y -onto-> x  <-> 
z : y -onto-> X ) )
1716exbidv 1637 . . . . . 6  |-  ( x  =  X  ->  ( E. z  z :
y -onto-> x  <->  E. z  z : y -onto-> X ) )
1815, 17orbi12d 692 . . . . 5  |-  ( x  =  X  ->  (
( x  =  (/)  \/ 
E. z  z : y -onto-> x )  <->  ( X  =  (/)  \/  E. z 
z : y -onto-> X ) ) )
19 foeq2 5652 . . . . . . 7  |-  ( y  =  Y  ->  (
z : y -onto-> X  <-> 
z : Y -onto-> X
) )
2019exbidv 1637 . . . . . 6  |-  ( y  =  Y  ->  ( E. z  z :
y -onto-> X  <->  E. z  z : Y -onto-> X ) )
2120orbi2d 684 . . . . 5  |-  ( y  =  Y  ->  (
( X  =  (/)  \/ 
E. z  z : y -onto-> X )  <->  ( X  =  (/)  \/  E. z 
z : Y -onto-> X
) ) )
22 df-wdom 7529 . . . . 5  |-  ~<_*  =  { <. x ,  y >.  |  ( x  =  (/)  \/  E. z  z : y
-onto-> x ) }
2318, 21, 22brabg 4476 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) )
2423expcom 426 . . 3  |-  ( Y  e.  _V  ->  ( X  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) ) ) )
254, 14, 24pm5.21ndd 345 . 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 set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   class class class wbr 4214   ran crn 4881   -onto->wfo 5454    ~<_* cwdom 7527
This theorem is referenced by:  brwdomi  7538  brwdomn0  7539  0wdom  7540  fowdom  7541  domwdom  7544  wdomnumr  7947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-dm 4890  df-rn 4891  df-fn 5459  df-fo 5462  df-wdom 7529
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