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Theorem domwdom 7989
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 2657 . . . . . . . 8  |-  ( X  =/=  (/)  <->  -.  X  =  (/) )
21biimpri 206 . . . . . . 7  |-  ( -.  X  =  (/)  ->  X  =/=  (/) )
32adantl 466 . . . . . 6  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  X  =/=  (/) )
4 reldom 7512 . . . . . . . . 9  |-  Rel  ~<_
54brrelexi 5032 . . . . . . . 8  |-  ( X  ~<_  Y  ->  X  e.  _V )
6 0sdomg 7636 . . . . . . . 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 7658 . . . . 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 5033 . . 3  |-  ( X  ~<_  Y  ->  Y  e.  _V )
16 brwdom 7982 . . 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 1374   E.wex 1591    e. wcel 1762    =/= wne 2655   _Vcvv 3106   (/)c0 3778   class class class wbr 4440   -onto->wfo 5577    ~<_ cdom 7504    ~< csdm 7505    ~<_* cwdom 7972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-wdom 7974
This theorem is referenced by:  wdomen1  7991  wdomen2  7992  wdom2d  7995  wdomima2g  8001  unxpwdom2  8003  unxpwdom  8004  harwdom  8005  wdomfil  8431  wdomnumr  8434  pwcdadom  8585  hsmexlem1  8795  hsmexlem4  8798
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