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Theorem wdomtr 7499
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 7490 . . . . 5  |-  Rel  ~<_*
21brrelex2i 4878 . . . 4  |-  ( Y  ~<_*  Z  ->  Z  e.  _V )
32adantl 453 . . 3  |-  ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  ->  Z  e.  _V )
4 0wdom 7494 . . . 4  |-  ( Z  e.  _V  ->  (/)  ~<_*  Z )
5 breq1 4175 . . . 4  |-  ( X  =  (/)  ->  ( X  ~<_*  Z 
<->  (/) 
~<_* 
Z ) )
64, 5syl5ibrcom 214 . . 3  |-  ( Z  e.  _V  ->  ( X  =  (/)  ->  X  ~<_*  Z ) )
73, 6syl 16 . 2  |-  ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  ->  ( X  =  (/)  ->  X  ~<_*  Z ) )
8 simpll 731 . . . . 5  |-  ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  /\  X  =/=  (/) )  ->  X  ~<_*  Y )
9 brwdomn0 7493 . . . . . 6  |-  ( X  =/=  (/)  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
109adantl 453 . . . . 5  |-  ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  /\  X  =/=  (/) )  ->  ( X  ~<_*  Y  <->  E. z  z : Y -onto-> X ) )
118, 10mpbid 202 . . . 4  |-  ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  /\  X  =/=  (/) )  ->  E. z 
z : Y -onto-> X
)
12 simpllr 736 . . . . . 6  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  Y  ~<_*  Z )
13 simplr 732 . . . . . . . 8  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  X  =/=  (/) )
14 dm0rn0 5045 . . . . . . . . . . . 12  |-  ( dom  z  =  (/)  <->  ran  z  =  (/) )
1514necon3bii 2599 . . . . . . . . . . 11  |-  ( dom  z  =/=  (/)  <->  ran  z  =/=  (/) )
1615a1i 11 . . . . . . . . . 10  |-  ( z : Y -onto-> X  -> 
( dom  z  =/=  (/)  <->  ran  z  =/=  (/) ) )
17 fof 5612 . . . . . . . . . . . 12  |-  ( z : Y -onto-> X  -> 
z : Y --> X )
18 fdm 5554 . . . . . . . . . . . 12  |-  ( z : Y --> X  ->  dom  z  =  Y
)
1917, 18syl 16 . . . . . . . . . . 11  |-  ( z : Y -onto-> X  ->  dom  z  =  Y
)
2019neeq1d 2580 . . . . . . . . . 10  |-  ( z : Y -onto-> X  -> 
( dom  z  =/=  (/)  <->  Y  =/=  (/) ) )
21 forn 5615 . . . . . . . . . . 11  |-  ( z : Y -onto-> X  ->  ran  z  =  X
)
2221neeq1d 2580 . . . . . . . . . 10  |-  ( z : Y -onto-> X  -> 
( ran  z  =/=  (/)  <->  X  =/=  (/) ) )
2316, 20, 223bitr3rd 276 . . . . . . . . 9  |-  ( z : Y -onto-> X  -> 
( X  =/=  (/)  <->  Y  =/=  (/) ) )
2423adantl 453 . . . . . . . 8  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  ( X  =/=  (/)  <->  Y  =/=  (/) ) )
2513, 24mpbid 202 . . . . . . 7  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  Y  =/=  (/) )
26 brwdomn0 7493 . . . . . . 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 202 . . . . 5  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  E. y 
y : Z -onto-> Y
)
29 vex 2919 . . . . . . . . . 10  |-  z  e. 
_V
30 vex 2919 . . . . . . . . . 10  |-  y  e. 
_V
3129, 30coex 5372 . . . . . . . . 9  |-  ( z  o.  y )  e. 
_V
32 foco 5622 . . . . . . . . 9  |-  ( ( z : Y -onto-> X  /\  y : Z -onto-> Y
)  ->  ( z  o.  y ) : Z -onto-> X )
33 fowdom 7495 . . . . . . . . 9  |-  ( ( ( z  o.  y
)  e.  _V  /\  ( z  o.  y
) : Z -onto-> X
)  ->  X  ~<_*  Z )
3431, 32, 33sylancr 645 . . . . . . . 8  |-  ( ( z : Y -onto-> X  /\  y : Z -onto-> Y
)  ->  X  ~<_*  Z )
3534adantl 453 . . . . . . 7  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  (
z : Y -onto-> X  /\  y : Z -onto-> Y
) )  ->  X  ~<_*  Z )
3635expr 599 . . . . . 6  |-  ( ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  /\  X  =/=  (/) )  /\  z : Y -onto-> X )  ->  (
y : Z -onto-> Y  ->  X  ~<_*  Z ) )
3736exlimdv 1643 . . . . 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 1645 . . 3  |-  ( ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  /\  X  =/=  (/) )  ->  X  ~<_*  Z )
4039ex 424 . 2  |-  ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  ->  ( X  =/=  (/)  ->  X  ~<_*  Z ) )
417, 40pm2.61dne 2644 1  |-  ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  ->  X  ~<_*  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916   (/)c0 3588   class class class wbr 4172   dom cdm 4837   ran crn 4838    o. ccom 4841   -->wf 5409   -onto->wfo 5411    ~<_* cwdom 7481
This theorem is referenced by:  wdomen1  7500  wdomen2  7501  wdom2d  7504  wdomima2g  7510  unxpwdom2  7512  unxpwdom  7513  harwdom  7514  pwcdadom  8052  hsmexlem1  8262  hsmexlem4  8265
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-wdom 7483
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