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Theorem rntpos 6022
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 2622 . . . . 5  |-  x  e. 
_V
21elrn 4678 . . . 4  |-  ( x  e.  ran tpos  F  <->  E. y 
ytpos  F x )
3 vex 2622 . . . . . . . . 9  |-  y  e. 
_V
43, 1breldm 4640 . . . . . . . 8  |-  ( ytpos 
F x  ->  y  e.  dom tpos  F )
5 dmtpos 6021 . . . . . . . . 9  |-  ( Rel 
dom  F  ->  dom tpos  F  =  `' dom  F )
65eleq2d 2157 . . . . . . . 8  |-  ( Rel 
dom  F  ->  ( y  e.  dom tpos  F  <->  y  e.  `' dom  F ) )
74, 6syl5ib 152 . . . . . . 7  |-  ( Rel 
dom  F  ->  ( ytpos 
F x  ->  y  e.  `' dom  F ) )
8 relcnv 4810 . . . . . . . 8  |-  Rel  `' dom  F
9 elrel 4540 . . . . . . . 8  |-  ( ( Rel  `' dom  F  /\  y  e.  `' dom  F )  ->  E. w E. z  y  =  <. w ,  z >.
)
108, 9mpan 415 . . . . . . 7  |-  ( y  e.  `' dom  F  ->  E. w E. z 
y  =  <. w ,  z >. )
117, 10syl6 33 . . . . . 6  |-  ( Rel 
dom  F  ->  ( ytpos 
F x  ->  E. w E. z  y  =  <. w ,  z >.
) )
12 breq1 3848 . . . . . . . . 9  |-  ( y  =  <. w ,  z
>.  ->  ( ytpos  F x 
<-> 
<. w ,  z >.tpos  F x ) )
13 vex 2622 . . . . . . . . . 10  |-  w  e. 
_V
14 vex 2622 . . . . . . . . . 10  |-  z  e. 
_V
15 brtposg 6019 . . . . . . . . . 10  |-  ( ( w  e.  _V  /\  z  e.  _V  /\  x  e.  _V )  ->  ( <. w ,  z >.tpos  F x  <->  <. z ,  w >. F x ) )
1613, 14, 1, 15mp3an 1273 . . . . . . . . 9  |-  ( <.
w ,  z >.tpos  F x  <->  <. z ,  w >. F x )
1712, 16syl6bb 194 . . . . . . . 8  |-  ( y  =  <. w ,  z
>.  ->  ( ytpos  F x 
<-> 
<. z ,  w >. F x ) )
1814, 13opex 4056 . . . . . . . . 9  |-  <. z ,  w >.  e.  _V
1918, 1brelrn 4668 . . . . . . . 8  |-  ( <.
z ,  w >. F x  ->  x  e.  ran  F )
2017, 19syl6bi 161 . . . . . . 7  |-  ( y  =  <. w ,  z
>.  ->  ( ytpos  F x  ->  x  e.  ran  F ) )
2120exlimivv 1824 . . . . . 6  |-  ( E. w E. z  y  =  <. w ,  z
>.  ->  ( ytpos  F x  ->  x  e.  ran  F ) )
2211, 21syli 37 . . . . 5  |-  ( Rel 
dom  F  ->  ( ytpos 
F x  ->  x  e.  ran  F ) )
2322exlimdv 1747 . . . 4  |-  ( Rel 
dom  F  ->  ( E. y  ytpos  F x  ->  x  e.  ran  F ) )
242, 23syl5bi 150 . . 3  |-  ( Rel 
dom  F  ->  ( x  e.  ran tpos  F  ->  x  e.  ran  F ) )
251elrn 4678 . . . 4  |-  ( x  e.  ran  F  <->  E. y 
y F x )
263, 1breldm 4640 . . . . . . 7  |-  ( y F x  ->  y  e.  dom  F )
27 elrel 4540 . . . . . . . 8  |-  ( ( Rel  dom  F  /\  y  e.  dom  F )  ->  E. z E. w  y  =  <. z ,  w >. )
2827ex 113 . . . . . . 7  |-  ( Rel 
dom  F  ->  ( y  e.  dom  F  ->  E. z E. w  y  =  <. z ,  w >. ) )
2926, 28syl5 32 . . . . . 6  |-  ( Rel 
dom  F  ->  ( y F x  ->  E. z E. w  y  =  <. z ,  w >. ) )
30 breq1 3848 . . . . . . . . 9  |-  ( y  =  <. z ,  w >.  ->  ( y F x  <->  <. z ,  w >. F x ) )
3130, 16syl6bbr 196 . . . . . . . 8  |-  ( y  =  <. z ,  w >.  ->  ( y F x  <->  <. w ,  z
>.tpos  F x ) )
3213, 14opex 4056 . . . . . . . . 9  |-  <. w ,  z >.  e.  _V
3332, 1brelrn 4668 . . . . . . . 8  |-  ( <.
w ,  z >.tpos  F x  ->  x  e. 
ran tpos  F )
3431, 33syl6bi 161 . . . . . . 7  |-  ( y  =  <. z ,  w >.  ->  ( y F x  ->  x  e.  ran tpos  F ) )
3534exlimivv 1824 . . . . . 6  |-  ( E. z E. w  y  =  <. z ,  w >.  ->  ( y F x  ->  x  e.  ran tpos  F ) )
3629, 35syli 37 . . . . 5  |-  ( Rel 
dom  F  ->  ( y F x  ->  x  e.  ran tpos  F ) )
3736exlimdv 1747 . . . 4  |-  ( Rel 
dom  F  ->  ( E. y  y F x  ->  x  e.  ran tpos  F ) )
3825, 37syl5bi 150 . . 3  |-  ( Rel 
dom  F  ->  ( x  e.  ran  F  ->  x  e.  ran tpos  F ) )
3924, 38impbid 127 . 2  |-  ( Rel 
dom  F  ->  ( x  e.  ran tpos  F  <->  x  e.  ran  F ) )
4039eqrdv 2086 1  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   <.cop 3449   class class class wbr 3845   `'ccnv 4437   dom cdm 4438   ran crn 4439   Rel wrel 4443  tpos ctpos 6009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023  df-tpos 6010
This theorem is referenced by:  tposfo2  6032
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