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Theorem rntpos 6236
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 2733 . . . . 5  |-  x  e. 
_V
21elrn 4854 . . . 4  |-  ( x  e.  ran tpos  F  <->  E. y 
ytpos  F x )
3 vex 2733 . . . . . . . . 9  |-  y  e. 
_V
43, 1breldm 4815 . . . . . . . 8  |-  ( ytpos 
F x  ->  y  e.  dom tpos  F )
5 dmtpos 6235 . . . . . . . . 9  |-  ( Rel 
dom  F  ->  dom tpos  F  =  `' dom  F )
65eleq2d 2240 . . . . . . . 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 4989 . . . . . . . 8  |-  Rel  `' dom  F
9 elrel 4713 . . . . . . . 8  |-  ( ( Rel  `' dom  F  /\  y  e.  `' dom  F )  ->  E. w E. z  y  =  <. w ,  z >.
)
108, 9mpan 422 . . . . . . 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 3992 . . . . . . . . 9  |-  ( y  =  <. w ,  z
>.  ->  ( ytpos  F x 
<-> 
<. w ,  z >.tpos  F x ) )
13 vex 2733 . . . . . . . . . 10  |-  w  e. 
_V
14 vex 2733 . . . . . . . . . 10  |-  z  e. 
_V
15 brtposg 6233 . . . . . . . . . 10  |-  ( ( w  e.  _V  /\  z  e.  _V  /\  x  e.  _V )  ->  ( <. w ,  z >.tpos  F x  <->  <. z ,  w >. F x ) )
1613, 14, 1, 15mp3an 1332 . . . . . . . . 9  |-  ( <.
w ,  z >.tpos  F x  <->  <. z ,  w >. F x )
1712, 16bitrdi 195 . . . . . . . 8  |-  ( y  =  <. w ,  z
>.  ->  ( ytpos  F x 
<-> 
<. z ,  w >. F x ) )
1814, 13opex 4214 . . . . . . . . 9  |-  <. z ,  w >.  e.  _V
1918, 1brelrn 4844 . . . . . . . 8  |-  ( <.
z ,  w >. F x  ->  x  e.  ran  F )
2017, 19syl6bi 162 . . . . . . 7  |-  ( y  =  <. w ,  z
>.  ->  ( ytpos  F x  ->  x  e.  ran  F ) )
2120exlimivv 1889 . . . . . 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 1812 . . . 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 4854 . . . 4  |-  ( x  e.  ran  F  <->  E. y 
y F x )
263, 1breldm 4815 . . . . . . 7  |-  ( y F x  ->  y  e.  dom  F )
27 elrel 4713 . . . . . . . 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 3992 . . . . . . . . 9  |-  ( y  =  <. z ,  w >.  ->  ( y F x  <->  <. z ,  w >. F x ) )
3130, 16bitr4di 197 . . . . . . . 8  |-  ( y  =  <. z ,  w >.  ->  ( y F x  <->  <. w ,  z
>.tpos  F x ) )
3213, 14opex 4214 . . . . . . . . 9  |-  <. w ,  z >.  e.  _V
3332, 1brelrn 4844 . . . . . . . 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 1889 . . . . . 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 1812 . . . 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 2168 1  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730   <.cop 3586   class class class wbr 3989   `'ccnv 4610   dom cdm 4611   ran crn 4612   Rel wrel 4616  tpos ctpos 6223
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-tpos 6224
This theorem is referenced by:  tposfo2  6246
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