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Theorem domwdom 8094
Description: Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Assertion
Ref Expression
domwdom  |-  ( X  ~<_  Y  ->  X  ~<_*  Y )

Proof of Theorem domwdom
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-ne 2626 . . . . . . . 8  |-  ( X  =/=  (/)  <->  -.  X  =  (/) )
21biimpri 210 . . . . . . 7  |-  ( -.  X  =  (/)  ->  X  =/=  (/) )
32adantl 468 . . . . . 6  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  X  =/=  (/) )
4 reldom 7580 . . . . . . . . 9  |-  Rel  ~<_
54brrelexi 4878 . . . . . . . 8  |-  ( X  ~<_  Y  ->  X  e.  _V )
6 0sdomg 7706 . . . . . . . 8  |-  ( X  e.  _V  ->  ( (/) 
~<  X  <->  X  =/=  (/) ) )
75, 6syl 17 . . . . . . 7  |-  ( X  ~<_  Y  ->  ( (/)  ~<  X  <->  X  =/=  (/) ) )
87adantr 467 . . . . . 6  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  -> 
( (/)  ~<  X  <->  X  =/=  (/) ) )
93, 8mpbird 236 . . . . 5  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  (/) 
~<  X )
10 simpl 459 . . . . 5  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  X  ~<_  Y )
11 fodomr 7728 . . . . 5  |-  ( (
(/)  ~<  X  /\  X  ~<_  Y )  ->  E. y 
y : Y -onto-> X
)
129, 10, 11syl2anc 667 . . . 4  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  E. y  y : Y -onto-> X )
1312ex 436 . . 3  |-  ( X  ~<_  Y  ->  ( -.  X  =  (/)  ->  E. y 
y : Y -onto-> X
) )
1413orrd 380 . 2  |-  ( X  ~<_  Y  ->  ( X  =  (/)  \/  E. y 
y : Y -onto-> X
) )
154brrelex2i 4879 . . 3  |-  ( X  ~<_  Y  ->  Y  e.  _V )
16 brwdom 8087 . . 3  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. y  y : Y -onto-> X ) ) )
1715, 16syl 17 . 2  |-  ( X  ~<_  Y  ->  ( X  ~<_*  Y  <-> 
( X  =  (/)  \/ 
E. y  y : Y -onto-> X ) ) )
1814, 17mpbird 236 1  |-  ( X  ~<_  Y  ->  X  ~<_*  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1446   E.wex 1665    e. wcel 1889    =/= wne 2624   _Vcvv 3047   (/)c0 3733   class class class wbr 4405   -onto->wfo 5583    ~<_ cdom 7572    ~< csdm 7573    ~<_* cwdom 8077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-wdom 8079
This theorem is referenced by:  wdomen1  8096  wdomen2  8097  wdom2d  8100  wdomima2g  8106  unxpwdom2  8108  unxpwdom  8109  harwdom  8110  wdomfil  8497  wdomnumr  8500  pwcdadom  8651  hsmexlem1  8861  hsmexlem4  8864
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