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Theorem wdomtr 7469
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 7460 . . . . 5  |-  Rel  ~<_*
21brrelex2i 4852 . . . 4  |-  ( Y  ~<_*  Z  ->  Z  e.  _V )
32adantl 453 . . 3  |-  ( ( X  ~<_*  Y  /\  Y  ~<_*  Z
)  ->  Z  e.  _V )
4 0wdom 7464 . . . 4  |-  ( Z  e.  _V  ->  (/)  ~<_*  Z )
5 breq1 4149 . . . 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 7463 . . . . . 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 5019 . . . . . . . . . . . 12  |-  ( dom  z  =  (/)  <->  ran  z  =  (/) )
1514necon3bii 2575 . . . . . . . . . . 11  |-  ( dom  z  =/=  (/)  <->  ran  z  =/=  (/) )
1615a1i 11 . . . . . . . . . 10  |-  ( z : Y -onto-> X  -> 
( dom  z  =/=  (/)  <->  ran  z  =/=  (/) ) )
17 fof 5586 . . . . . . . . . . . 12  |-  ( z : Y -onto-> X  -> 
z : Y --> X )
18 fdm 5528 . . . . . . . . . . . 12  |-  ( z : Y --> X  ->  dom  z  =  Y
)
1917, 18syl 16 . . . . . . . . . . 11  |-  ( z : Y -onto-> X  ->  dom  z  =  Y
)
2019neeq1d 2556 . . . . . . . . . 10  |-  ( z : Y -onto-> X  -> 
( dom  z  =/=  (/)  <->  Y  =/=  (/) ) )
21 forn 5589 . . . . . . . . . . 11  |-  ( z : Y -onto-> X  ->  ran  z  =  X
)
2221neeq1d 2556 . . . . . . . . . 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 7463 . . . . . . 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 2895 . . . . . . . . . 10  |-  z  e. 
_V
30 vex 2895 . . . . . . . . . 10  |-  y  e. 
_V
3129, 30coex 5346 . . . . . . . . 9  |-  ( z  o.  y )  e. 
_V
32 foco 5596 . . . . . . . . 9  |-  ( ( z : Y -onto-> X  /\  y : Z -onto-> Y
)  ->  ( z  o.  y ) : Z -onto-> X )
33 fowdom 7465 . . . . . . . . 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 2620 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 1717    =/= wne 2543   _Vcvv 2892   (/)c0 3564   class class class wbr 4146   dom cdm 4811   ran crn 4812    o. ccom 4815   -->wf 5383   -onto->wfo 5385    ~<_* cwdom 7451
This theorem is referenced by:  wdomen1  7470  wdomen2  7471  wdom2d  7474  wdomima2g  7480  unxpwdom2  7482  unxpwdom  7483  harwdom  7484  pwcdadom  8022  hsmexlem1  8232  hsmexlem4  8235
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 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-fun 5389  df-fn 5390  df-f 5391  df-fo 5393  df-wdom 7453
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