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Theorem domwdom 8003
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 2640 . . . . . . . 8  |-  ( X  =/=  (/)  <->  -.  X  =  (/) )
21biimpri 206 . . . . . . 7  |-  ( -.  X  =  (/)  ->  X  =/=  (/) )
32adantl 466 . . . . . 6  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  X  =/=  (/) )
4 reldom 7524 . . . . . . . . 9  |-  Rel  ~<_
54brrelexi 5030 . . . . . . . 8  |-  ( X  ~<_  Y  ->  X  e.  _V )
6 0sdomg 7648 . . . . . . . 8  |-  ( X  e.  _V  ->  ( (/) 
~<  X  <->  X  =/=  (/) ) )
75, 6syl 16 . . . . . . 7  |-  ( X  ~<_  Y  ->  ( (/)  ~<  X  <->  X  =/=  (/) ) )
87adantr 465 . . . . . 6  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  -> 
( (/)  ~<  X  <->  X  =/=  (/) ) )
93, 8mpbird 232 . . . . 5  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  (/) 
~<  X )
10 simpl 457 . . . . 5  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  X  ~<_  Y )
11 fodomr 7670 . . . . 5  |-  ( (
(/)  ~<  X  /\  X  ~<_  Y )  ->  E. y 
y : Y -onto-> X
)
129, 10, 11syl2anc 661 . . . 4  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  E. y  y : Y -onto-> X )
1312ex 434 . . 3  |-  ( X  ~<_  Y  ->  ( -.  X  =  (/)  ->  E. y 
y : Y -onto-> X
) )
1413orrd 378 . 2  |-  ( X  ~<_  Y  ->  ( X  =  (/)  \/  E. y 
y : Y -onto-> X
) )
154brrelex2i 5031 . . 3  |-  ( X  ~<_  Y  ->  Y  e.  _V )
16 brwdom 7996 . . 3  |-  ( Y  e.  _V  ->  ( X  ~<_*  Y  <->  ( X  =  (/)  \/  E. y  y : Y -onto-> X ) ) )
1715, 16syl 16 . 2  |-  ( X  ~<_  Y  ->  ( X  ~<_*  Y  <-> 
( X  =  (/)  \/ 
E. y  y : Y -onto-> X ) ) )
1814, 17mpbird 232 1  |-  ( X  ~<_  Y  ->  X  ~<_*  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804    =/= wne 2638   _Vcvv 3095   (/)c0 3770   class class class wbr 4437   -onto->wfo 5576    ~<_ cdom 7516    ~< csdm 7517    ~<_* cwdom 7986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-wdom 7988
This theorem is referenced by:  wdomen1  8005  wdomen2  8006  wdom2d  8009  wdomima2g  8015  unxpwdom2  8017  unxpwdom  8018  harwdom  8019  wdomfil  8445  wdomnumr  8448  pwcdadom  8599  hsmexlem1  8809  hsmexlem4  8812
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