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Theorem domwdom 7477
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 2554 . . . . . . . 8  |-  ( X  =/=  (/)  <->  -.  X  =  (/) )
21biimpri 198 . . . . . . 7  |-  ( -.  X  =  (/)  ->  X  =/=  (/) )
32adantl 453 . . . . . 6  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  X  =/=  (/) )
4 reldom 7053 . . . . . . . . 9  |-  Rel  ~<_
54brrelexi 4860 . . . . . . . 8  |-  ( X  ~<_  Y  ->  X  e.  _V )
6 0sdomg 7174 . . . . . . . 8  |-  ( X  e.  _V  ->  ( (/) 
~<  X  <->  X  =/=  (/) ) )
75, 6syl 16 . . . . . . 7  |-  ( X  ~<_  Y  ->  ( (/)  ~<  X  <->  X  =/=  (/) ) )
87adantr 452 . . . . . 6  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  -> 
( (/)  ~<  X  <->  X  =/=  (/) ) )
93, 8mpbird 224 . . . . 5  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  (/) 
~<  X )
10 simpl 444 . . . . 5  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  X  ~<_  Y )
11 fodomr 7196 . . . . 5  |-  ( (
(/)  ~<  X  /\  X  ~<_  Y )  ->  E. y 
y : Y -onto-> X
)
129, 10, 11syl2anc 643 . . . 4  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  E. y  y : Y -onto-> X )
1312ex 424 . . 3  |-  ( X  ~<_  Y  ->  ( -.  X  =  (/)  ->  E. y 
y : Y -onto-> X
) )
1413orrd 368 . 2  |-  ( X  ~<_  Y  ->  ( X  =  (/)  \/  E. y 
y : Y -onto-> X
) )
154brrelex2i 4861 . . 3  |-  ( X  ~<_  Y  ->  Y  e.  _V )
16 brwdom 7470 . . 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 224 1  |-  ( X  ~<_  Y  ->  X  ~<_*  Y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2552   _Vcvv 2901   (/)c0 3573   class class class wbr 4155   -onto->wfo 5394    ~<_ cdom 7045    ~< csdm 7046    ~<_* cwdom 7460
This theorem is referenced by:  wdomen1  7479  wdomen2  7480  wdom2d  7483  wdomima2g  7489  unxpwdom2  7491  unxpwdom  7492  harwdom  7493  wdomfil  7877  wdomnumr  7880  pwcdadom  8031  hsmexlem1  8241  hsmexlem4  8244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-wdom 7462
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