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Theorem rntpos 6263
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 2804 . . . . 5  |-  z  e. 
_V
21elrn 4935 . . . 4  |-  ( z  e.  ran tpos  F  <->  E. w  wtpos  F z )
3 vex 2804 . . . . . . . . 9  |-  w  e. 
_V
43, 1breldm 4899 . . . . . . . 8  |-  ( wtpos 
F z  ->  w  e.  dom tpos  F )
5 dmtpos 6262 . . . . . . . . 9  |-  ( Rel 
dom  F  ->  dom tpos  F  =  `' dom  F )
65eleq2d 2363 . . . . . . . 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 5067 . . . . . . . 8  |-  Rel  `' dom  F
9 elrel 4805 . . . . . . . 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 4042 . . . . . . . . 9  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  <->  <. x ,  y
>.tpos  F z ) )
13 brtpos 6259 . . . . . . . . . 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 4253 . . . . . . . . 9  |-  <. y ,  x >.  e.  _V
1716, 1brelrn 4925 . . . . . . . 8  |-  ( <.
y ,  x >. F z  ->  z  e.  ran  F )
1815, 17syl6bi 219 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  ->  z  e.  ran  F ) )
1918exlimivv 1625 . . . . . 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 1626 . . . 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 4935 . . . 4  |-  ( z  e.  ran  F  <->  E. w  w F z )
243, 1breldm 4899 . . . . . . 7  |-  ( w F z  ->  w  e.  dom  F )
25 elrel 4805 . . . . . . . 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 4042 . . . . . . . . 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 4253 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
3130, 1brelrn 4925 . . . . . . . 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 1625 . . . . . 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 1626 . . . 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 2294 1  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   ran crn 4706   Rel wrel 4710  tpos ctpos 6249
This theorem is referenced by:  tposfo2  6273  oppchofcl  14050  oyoncl  14060  dualalg  25885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-tpos 6250
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