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Theorem rntpos 6120
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 2661 . . . . 5  |-  x  e. 
_V
21elrn 4750 . . . 4  |-  ( x  e.  ran tpos  F  <->  E. y 
ytpos  F x )
3 vex 2661 . . . . . . . . 9  |-  y  e. 
_V
43, 1breldm 4711 . . . . . . . 8  |-  ( ytpos 
F x  ->  y  e.  dom tpos  F )
5 dmtpos 6119 . . . . . . . . 9  |-  ( Rel 
dom  F  ->  dom tpos  F  =  `' dom  F )
65eleq2d 2185 . . . . . . . 8  |-  ( Rel 
dom  F  ->  ( y  e.  dom tpos  F  <->  y  e.  `' dom  F ) )
74, 6syl5ib 153 . . . . . . 7  |-  ( Rel 
dom  F  ->  ( ytpos 
F x  ->  y  e.  `' dom  F ) )
8 relcnv 4885 . . . . . . . 8  |-  Rel  `' dom  F
9 elrel 4609 . . . . . . . 8  |-  ( ( Rel  `' dom  F  /\  y  e.  `' dom  F )  ->  E. w E. z  y  =  <. w ,  z >.
)
108, 9mpan 418 . . . . . . 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 3900 . . . . . . . . 9  |-  ( y  =  <. w ,  z
>.  ->  ( ytpos  F x 
<-> 
<. w ,  z >.tpos  F x ) )
13 vex 2661 . . . . . . . . . 10  |-  w  e. 
_V
14 vex 2661 . . . . . . . . . 10  |-  z  e. 
_V
15 brtposg 6117 . . . . . . . . . 10  |-  ( ( w  e.  _V  /\  z  e.  _V  /\  x  e.  _V )  ->  ( <. w ,  z >.tpos  F x  <->  <. z ,  w >. F x ) )
1613, 14, 1, 15mp3an 1298 . . . . . . . . 9  |-  ( <.
w ,  z >.tpos  F x  <->  <. z ,  w >. F x )
1712, 16syl6bb 195 . . . . . . . 8  |-  ( y  =  <. w ,  z
>.  ->  ( ytpos  F x 
<-> 
<. z ,  w >. F x ) )
1814, 13opex 4119 . . . . . . . . 9  |-  <. z ,  w >.  e.  _V
1918, 1brelrn 4740 . . . . . . . 8  |-  ( <.
z ,  w >. F x  ->  x  e.  ran  F )
2017, 19syl6bi 162 . . . . . . 7  |-  ( y  =  <. w ,  z
>.  ->  ( ytpos  F x  ->  x  e.  ran  F ) )
2120exlimivv 1850 . . . . . 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 1773 . . . 4  |-  ( Rel 
dom  F  ->  ( E. y  ytpos  F x  ->  x  e.  ran  F ) )
242, 23syl5bi 151 . . 3  |-  ( Rel 
dom  F  ->  ( x  e.  ran tpos  F  ->  x  e.  ran  F ) )
251elrn 4750 . . . 4  |-  ( x  e.  ran  F  <->  E. y 
y F x )
263, 1breldm 4711 . . . . . . 7  |-  ( y F x  ->  y  e.  dom  F )
27 elrel 4609 . . . . . . . 8  |-  ( ( Rel  dom  F  /\  y  e.  dom  F )  ->  E. z E. w  y  =  <. z ,  w >. )
2827ex 114 . . . . . . 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 3900 . . . . . . . . 9  |-  ( y  =  <. z ,  w >.  ->  ( y F x  <->  <. z ,  w >. F x ) )
3130, 16syl6bbr 197 . . . . . . . 8  |-  ( y  =  <. z ,  w >.  ->  ( y F x  <->  <. w ,  z
>.tpos  F x ) )
3213, 14opex 4119 . . . . . . . . 9  |-  <. w ,  z >.  e.  _V
3332, 1brelrn 4740 . . . . . . . 8  |-  ( <.
w ,  z >.tpos  F x  ->  x  e. 
ran tpos  F )
3431, 33syl6bi 162 . . . . . . 7  |-  ( y  =  <. z ,  w >.  ->  ( y F x  ->  x  e.  ran tpos  F ) )
3534exlimivv 1850 . . . . . 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 1773 . . . 4  |-  ( Rel 
dom  F  ->  ( E. y  y F x  ->  x  e.  ran tpos  F ) )
3825, 37syl5bi 151 . . 3  |-  ( Rel 
dom  F  ->  ( x  e.  ran  F  ->  x  e.  ran tpos  F ) )
3924, 38impbid 128 . 2  |-  ( Rel 
dom  F  ->  ( x  e.  ran tpos  F  <->  x  e.  ran  F ) )
4039eqrdv 2113 1  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   _Vcvv 2658   <.cop 3498   class class class wbr 3897   `'ccnv 4506   dom cdm 4507   ran crn 4508   Rel wrel 4512  tpos ctpos 6107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099  df-tpos 6108
This theorem is referenced by:  tposfo2  6130
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