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Theorem rntpos 6484
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 2951 . . . . 5  |-  z  e. 
_V
21elrn 5102 . . . 4  |-  ( z  e.  ran tpos  F  <->  E. w  wtpos  F z )
3 vex 2951 . . . . . . . . 9  |-  w  e. 
_V
43, 1breldm 5066 . . . . . . . 8  |-  ( wtpos 
F z  ->  w  e.  dom tpos  F )
5 dmtpos 6483 . . . . . . . . 9  |-  ( Rel 
dom  F  ->  dom tpos  F  =  `' dom  F )
65eleq2d 2502 . . . . . . . 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 5234 . . . . . . . 8  |-  Rel  `' dom  F
9 elrel 4970 . . . . . . . 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 4207 . . . . . . . . 9  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  <->  <. x ,  y
>.tpos  F z ) )
13 brtpos 6480 . . . . . . . . . 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 4419 . . . . . . . . 9  |-  <. y ,  x >.  e.  _V
1716, 1brelrn 5092 . . . . . . . 8  |-  ( <.
y ,  x >. F z  ->  z  e.  ran  F )
1815, 17syl6bi 220 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  ->  z  e.  ran  F ) )
1918exlimivv 1645 . . . . . 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 1646 . . . 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 5102 . . . 4  |-  ( z  e.  ran  F  <->  E. w  w F z )
243, 1breldm 5066 . . . . . . 7  |-  ( w F z  ->  w  e.  dom  F )
25 elrel 4970 . . . . . . . 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 4207 . . . . . . . . 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 4419 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
3130, 1brelrn 5092 . . . . . . . 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 1645 . . . . . 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 1646 . . . 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 2433 1  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   ran crn 4871   Rel wrel 4875  tpos ctpos 6470
This theorem is referenced by:  tposfo2  6494  oppchofcl  14349  oyoncl  14359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-tpos 6471
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