MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rntpos Unicode version

Theorem rntpos 6429
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 2903 . . . . 5  |-  z  e. 
_V
21elrn 5051 . . . 4  |-  ( z  e.  ran tpos  F  <->  E. w  wtpos  F z )
3 vex 2903 . . . . . . . . 9  |-  w  e. 
_V
43, 1breldm 5015 . . . . . . . 8  |-  ( wtpos 
F z  ->  w  e.  dom tpos  F )
5 dmtpos 6428 . . . . . . . . 9  |-  ( Rel 
dom  F  ->  dom tpos  F  =  `' dom  F )
65eleq2d 2455 . . . . . . . 8  |-  ( Rel 
dom  F  ->  ( w  e.  dom tpos  F  <->  w  e.  `' dom  F ) )
74, 6syl5ib 211 . . . . . . 7  |-  ( Rel 
dom  F  ->  ( wtpos 
F z  ->  w  e.  `' dom  F ) )
8 relcnv 5183 . . . . . . . 8  |-  Rel  `' dom  F
9 elrel 4919 . . . . . . . 8  |-  ( ( Rel  `' dom  F  /\  w  e.  `' dom  F )  ->  E. x E. y  w  =  <. x ,  y >.
)
108, 9mpan 652 . . . . . . 7  |-  ( w  e.  `' dom  F  ->  E. x E. y  w  =  <. x ,  y >. )
117, 10syl6 31 . . . . . 6  |-  ( Rel 
dom  F  ->  ( wtpos 
F z  ->  E. x E. y  w  =  <. x ,  y >.
) )
12 breq1 4157 . . . . . . . . 9  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  <->  <. x ,  y
>.tpos  F z ) )
13 brtpos 6425 . . . . . . . . . 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 253 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  <->  <. y ,  x >. F z ) )
16 opex 4369 . . . . . . . . 9  |-  <. y ,  x >.  e.  _V
1716, 1brelrn 5041 . . . . . . . 8  |-  ( <.
y ,  x >. F z  ->  z  e.  ran  F )
1815, 17syl6bi 220 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  ->  z  e.  ran  F ) )
1918exlimivv 1642 . . . . . 6  |-  ( E. x E. y  w  =  <. x ,  y
>.  ->  ( wtpos  F
z  ->  z  e.  ran  F ) )
2011, 19syli 35 . . . . 5  |-  ( Rel 
dom  F  ->  ( wtpos 
F z  ->  z  e.  ran  F ) )
2120exlimdv 1643 . . . 4  |-  ( Rel 
dom  F  ->  ( E. w  wtpos  F z  ->  z  e.  ran  F ) )
222, 21syl5bi 209 . . 3  |-  ( Rel 
dom  F  ->  ( z  e.  ran tpos  F  ->  z  e.  ran  F ) )
231elrn 5051 . . . 4  |-  ( z  e.  ran  F  <->  E. w  w F z )
243, 1breldm 5015 . . . . . . 7  |-  ( w F z  ->  w  e.  dom  F )
25 elrel 4919 . . . . . . . 8  |-  ( ( Rel  dom  F  /\  w  e.  dom  F )  ->  E. y E. x  w  =  <. y ,  x >. )
2625ex 424 . . . . . . 7  |-  ( Rel 
dom  F  ->  ( w  e.  dom  F  ->  E. y E. x  w  =  <. y ,  x >. ) )
2724, 26syl5 30 . . . . . 6  |-  ( Rel 
dom  F  ->  ( w F z  ->  E. y E. x  w  =  <. y ,  x >. ) )
28 breq1 4157 . . . . . . . . 9  |-  ( w  =  <. y ,  x >.  ->  ( w F z  <->  <. y ,  x >. F z ) )
2928, 14syl6bbr 255 . . . . . . . 8  |-  ( w  =  <. y ,  x >.  ->  ( w F z  <->  <. x ,  y
>.tpos  F z ) )
30 opex 4369 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
3130, 1brelrn 5041 . . . . . . . 8  |-  ( <.
x ,  y >.tpos  F z  ->  z  e.  ran tpos  F )
3229, 31syl6bi 220 . . . . . . 7  |-  ( w  =  <. y ,  x >.  ->  ( w F z  ->  z  e.  ran tpos  F ) )
3332exlimivv 1642 . . . . . 6  |-  ( E. y E. x  w  =  <. y ,  x >.  ->  ( w F z  ->  z  e.  ran tpos  F ) )
3427, 33syli 35 . . . . 5  |-  ( Rel 
dom  F  ->  ( w F z  ->  z  e.  ran tpos  F ) )
3534exlimdv 1643 . . . 4  |-  ( Rel 
dom  F  ->  ( E. w  w F z  ->  z  e.  ran tpos  F ) )
3623, 35syl5bi 209 . . 3  |-  ( Rel 
dom  F  ->  ( z  e.  ran  F  -> 
z  e.  ran tpos  F ) )
3722, 36impbid 184 . 2  |-  ( Rel 
dom  F  ->  ( z  e.  ran tpos  F  <->  z  e.  ran  F ) )
3837eqrdv 2386 1  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2900   <.cop 3761   class class class wbr 4154   `'ccnv 4818   dom cdm 4819   ran crn 4820   Rel wrel 4824  tpos ctpos 6415
This theorem is referenced by:  tposfo2  6439  oppchofcl  14285  oyoncl  14295
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403  df-tpos 6416
  Copyright terms: Public domain W3C validator