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Theorem rntpos 6247
Description: The range of tpos  F when  dom  F is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
rntpos  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )

Proof of Theorem rntpos
Dummy variables  x  y  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . 5  |-  z  e. 
_V
21elrn 4919 . . . 4  |-  ( z  e.  ran tpos  F  <->  E. w  wtpos  F z )
3 vex 2791 . . . . . . . . 9  |-  w  e. 
_V
43, 1breldm 4883 . . . . . . . 8  |-  ( wtpos 
F z  ->  w  e.  dom tpos  F )
5 dmtpos 6246 . . . . . . . . 9  |-  ( Rel 
dom  F  ->  dom tpos  F  =  `' dom  F )
65eleq2d 2350 . . . . . . . 8  |-  ( Rel 
dom  F  ->  ( w  e.  dom tpos  F  <->  w  e.  `' dom  F ) )
74, 6syl5ib 210 . . . . . . 7  |-  ( Rel 
dom  F  ->  ( wtpos 
F z  ->  w  e.  `' dom  F ) )
8 relcnv 5051 . . . . . . . 8  |-  Rel  `' dom  F
9 elrel 4789 . . . . . . . 8  |-  ( ( Rel  `' dom  F  /\  w  e.  `' dom  F )  ->  E. x E. y  w  =  <. x ,  y >.
)
108, 9mpan 651 . . . . . . 7  |-  ( w  e.  `' dom  F  ->  E. x E. y  w  =  <. x ,  y >. )
117, 10syl6 29 . . . . . 6  |-  ( Rel 
dom  F  ->  ( wtpos 
F z  ->  E. x E. y  w  =  <. x ,  y >.
) )
12 breq1 4026 . . . . . . . . 9  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  <->  <. x ,  y
>.tpos  F z ) )
13 brtpos 6243 . . . . . . . . . 10  |-  ( z  e.  _V  ->  ( <. x ,  y >.tpos  F z  <->  <. y ,  x >. F z ) )
141, 13ax-mp 8 . . . . . . . . 9  |-  ( <.
x ,  y >.tpos  F z  <->  <. y ,  x >. F z )
1512, 14syl6bb 252 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  <->  <. y ,  x >. F z ) )
16 opex 4237 . . . . . . . . 9  |-  <. y ,  x >.  e.  _V
1716, 1brelrn 4909 . . . . . . . 8  |-  ( <.
y ,  x >. F z  ->  z  e.  ran  F )
1815, 17syl6bi 219 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  ->  z  e.  ran  F ) )
1918exlimivv 1667 . . . . . 6  |-  ( E. x E. y  w  =  <. x ,  y
>.  ->  ( wtpos  F
z  ->  z  e.  ran  F ) )
2011, 19syli 33 . . . . 5  |-  ( Rel 
dom  F  ->  ( wtpos 
F z  ->  z  e.  ran  F ) )
2120exlimdv 1664 . . . 4  |-  ( Rel 
dom  F  ->  ( E. w  wtpos  F z  ->  z  e.  ran  F ) )
222, 21syl5bi 208 . . 3  |-  ( Rel 
dom  F  ->  ( z  e.  ran tpos  F  ->  z  e.  ran  F ) )
231elrn 4919 . . . 4  |-  ( z  e.  ran  F  <->  E. w  w F z )
243, 1breldm 4883 . . . . . . 7  |-  ( w F z  ->  w  e.  dom  F )
25 elrel 4789 . . . . . . . 8  |-  ( ( Rel  dom  F  /\  w  e.  dom  F )  ->  E. y E. x  w  =  <. y ,  x >. )
2625ex 423 . . . . . . 7  |-  ( Rel 
dom  F  ->  ( w  e.  dom  F  ->  E. y E. x  w  =  <. y ,  x >. ) )
2724, 26syl5 28 . . . . . 6  |-  ( Rel 
dom  F  ->  ( w F z  ->  E. y E. x  w  =  <. y ,  x >. ) )
28 breq1 4026 . . . . . . . . 9  |-  ( w  =  <. y ,  x >.  ->  ( w F z  <->  <. y ,  x >. F z ) )
2928, 14syl6bbr 254 . . . . . . . 8  |-  ( w  =  <. y ,  x >.  ->  ( w F z  <->  <. x ,  y
>.tpos  F z ) )
30 opex 4237 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
3130, 1brelrn 4909 . . . . . . . 8  |-  ( <.
x ,  y >.tpos  F z  ->  z  e.  ran tpos  F )
3229, 31syl6bi 219 . . . . . . 7  |-  ( w  =  <. y ,  x >.  ->  ( w F z  ->  z  e.  ran tpos  F ) )
3332exlimivv 1667 . . . . . 6  |-  ( E. y E. x  w  =  <. y ,  x >.  ->  ( w F z  ->  z  e.  ran tpos  F ) )
3427, 33syli 33 . . . . 5  |-  ( Rel 
dom  F  ->  ( w F z  ->  z  e.  ran tpos  F ) )
3534exlimdv 1664 . . . 4  |-  ( Rel 
dom  F  ->  ( E. w  w F z  ->  z  e.  ran tpos  F ) )
3623, 35syl5bi 208 . . 3  |-  ( Rel 
dom  F  ->  ( z  e.  ran  F  -> 
z  e.  ran tpos  F ) )
3722, 36impbid 183 . 2  |-  ( Rel 
dom  F  ->  ( z  e.  ran tpos  F  <->  z  e.  ran  F ) )
3837eqrdv 2281 1  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ran crn 4690   Rel wrel 4694  tpos ctpos 6233
This theorem is referenced by:  tposfo2  6257  oppchofcl  14034  oyoncl  14044  dualalg  25782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-tpos 6234
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