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Theorem wdomtr 8004
Description: Transitivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
wdomtr  |-  ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  ->  X  ~<_*  Z )

Proof of Theorem wdomtr
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relwdom 7995 . . . . 5  |-  Rel  ~<_*
21brrelex2i 5031 . . . 4  |-  ( Y  ~<_*  Z  ->  Z  e.  _V )
32adantl 466 . . 3  |-  ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  ->  Z  e.  _V )
4 0wdom 7999 . . . 4  |-  ( Z  e.  _V  ->  (/)  ~<_*  Z )
5 breq1 4440 . . . 4  |-  ( X  =  (/)  ->  ( X  ~<_*  Z 
<->  (/) 
~<_* 
Z ) )
64, 5syl5ibrcom 222 . . 3  |-  ( Z  e.  _V  ->  ( X  =  (/)  ->  X  ~<_*  Z ) )
73, 6syl 16 . 2  |-  ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  ->  ( X  =  (/)  ->  X  ~<_*  Z ) )
8 simpll 753 . . . . 5  |-  ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  /\  X  =/=  (/) )  ->  X  ~<_*  Y )
9 brwdomn0 7998 . . . . . 6  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
109adantl 466 . . . . 5  |-  ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  /\  X  =/=  (/) )  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
118, 10mpbid 210 . . . 4  |-  ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  /\  X  =/=  (/) )  ->  E. z 
z : Y -onto-> X
)
12 simpllr 760 . . . . . 6  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  Y  ~<_*  Z )
13 simplr 755 . . . . . . . 8  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  X  =/=  (/) )
14 dm0rn0 5209 . . . . . . . . . . . 12  |-  ( dom  z  =  (/)  <->  ran  z  =  (/) )
1514necon3bii 2711 . . . . . . . . . . 11  |-  ( dom  z  =/=  (/)  <->  ran  z  =/=  (/) )
1615a1i 11 . . . . . . . . . 10  |-  ( z : Y -onto-> X  -> 
( dom  z  =/=  (/)  <->  ran  z  =/=  (/) ) )
17 fof 5785 . . . . . . . . . . . 12  |-  ( z : Y -onto-> X  -> 
z : Y --> X )
18 fdm 5725 . . . . . . . . . . . 12  |-  ( z : Y --> X  ->  dom  z  =  Y
)
1917, 18syl 16 . . . . . . . . . . 11  |-  ( z : Y -onto-> X  ->  dom  z  =  Y
)
2019neeq1d 2720 . . . . . . . . . 10  |-  ( z : Y -onto-> X  -> 
( dom  z  =/=  (/)  <->  Y  =/=  (/) ) )
21 forn 5788 . . . . . . . . . . 11  |-  ( z : Y -onto-> X  ->  ran  z  =  X
)
2221neeq1d 2720 . . . . . . . . . 10  |-  ( z : Y -onto-> X  -> 
( ran  z  =/=  (/)  <->  X  =/=  (/) ) )
2316, 20, 223bitr3rd 284 . . . . . . . . 9  |-  ( z : Y -onto-> X  -> 
( X  =/=  (/)  <->  Y  =/=  (/) ) )
2423adantl 466 . . . . . . . 8  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  ( X  =/=  (/)  <->  Y  =/=  (/) ) )
2513, 24mpbid 210 . . . . . . 7  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  Y  =/=  (/) )
26 brwdomn0 7998 . . . . . . 7  |-  ( Y  =/=  (/)  ->  ( Y  ~<_*  Z  <->  E. y  y : Z -onto-> Y ) )
2725, 26syl 16 . . . . . 6  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  ( Y  ~<_*  Z  <->  E. y  y : Z -onto-> Y ) )
2812, 27mpbid 210 . . . . 5  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  E. y 
y : Z -onto-> Y
)
29 vex 3098 . . . . . . . . . 10  |-  z  e. 
_V
30 vex 3098 . . . . . . . . . 10  |-  y  e. 
_V
3129, 30coex 6737 . . . . . . . . 9  |-  ( z  o.  y )  e. 
_V
32 foco 5795 . . . . . . . . 9  |-  ( ( z : Y -onto-> X  /\  y : Z -onto-> Y
)  ->  ( z  o.  y ) : Z -onto-> X )
33 fowdom 8000 . . . . . . . . 9  |-  ( ( ( z  o.  y
)  e.  _V  /\  ( z  o.  y
) : Z -onto-> X
)  ->  X  ~<_*  Z )
3431, 32, 33sylancr 663 . . . . . . . 8  |-  ( ( z : Y -onto-> X  /\  y : Z -onto-> Y
)  ->  X  ~<_*  Z )
3534adantl 466 . . . . . . 7  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  (
z : Y -onto-> X  /\  y : Z -onto-> Y
) )  ->  X  ~<_*  Z )
3635expr 615 . . . . . 6  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  (
y : Z -onto-> Y  ->  X  ~<_*  Z ) )
3736exlimdv 1711 . . . . 5  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  ( E. y  y : Z -onto-> Y  ->  X  ~<_*  Z
) )
3828, 37mpd 15 . . . 4  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  X  ~<_*  Z )
3911, 38exlimddv 1713 . . 3  |-  ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  /\  X  =/=  (/) )  ->  X  ~<_*  Z )
4039ex 434 . 2  |-  ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  ->  ( X  =/=  (/)  ->  X  ~<_*  Z ) )
417, 40pm2.61dne 2760 1  |-  ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  ->  X  ~<_*  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804    =/= wne 2638   _Vcvv 3095   (/)c0 3770   class class class wbr 4437   dom cdm 4989   ran crn 4990    o. ccom 4993   -->wf 5574   -onto->wfo 5576    ~<_* 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-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-wdom 7988
This theorem is referenced by:  wdomen1  8005  wdomen2  8006  wdom2d  8009  wdomima2g  8015  unxpwdom2  8017  unxpwdom  8018  harwdom  8019  pwcdadom  8599  hsmexlem1  8809  hsmexlem4  8812
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