MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  domwdom Structured version   Unicode version

Theorem domwdom 8097
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 2621 . . . . . . . 8  |-  ( X  =/=  (/)  <->  -.  X  =  (/) )
21biimpri 210 . . . . . . 7  |-  ( -.  X  =  (/)  ->  X  =/=  (/) )
32adantl 468 . . . . . 6  |-  ( ( X  ~<_  Y  /\  -.  X  =  (/) )  ->  X  =/=  (/) )
4 reldom 7585 . . . . . . . . 9  |-  Rel  ~<_
54brrelexi 4893 . . . . . . . 8  |-  ( X  ~<_  Y  ->  X  e.  _V )
6 0sdomg 7709 . . . . . . . 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 7731 . . . . 5  |-  ( (
(/)  ~<  X  /\  X  ~<_  Y )  ->  E. y 
y : Y -onto-> X
)
129, 10, 11syl2anc 666 . . . 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 4894 . . 3  |-  ( X  ~<_  Y  ->  Y  e.  _V )
16 brwdom 8090 . . 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 1438   E.wex 1660    e. wcel 1869    =/= wne 2619   _Vcvv 3082   (/)c0 3763   class class class wbr 4422   -onto->wfo 5598    ~<_ cdom 7577    ~< csdm 7578    ~<_* cwdom 8080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4545  ax-nul 4554  ax-pow 4601  ax-pr 4659  ax-un 6596
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-nul 3764  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4219  df-br 4423  df-opab 4482  df-mpt 4483  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-er 7373  df-en 7580  df-dom 7581  df-sdom 7582  df-wdom 8082
This theorem is referenced by:  wdomen1  8099  wdomen2  8100  wdom2d  8103  wdomima2g  8109  unxpwdom2  8111  unxpwdom  8112  harwdom  8113  wdomfil  8498  wdomnumr  8501  pwcdadom  8652  hsmexlem1  8862  hsmexlem4  8865
  Copyright terms: Public domain W3C validator