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Theorem rntpos 5903
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 2577 . . . . 5  |-  x  e. 
_V
21elrn 4605 . . . 4  |-  ( x  e.  ran tpos  F  <->  E. y 
ytpos  F x )
3 vex 2577 . . . . . . . . 9  |-  y  e. 
_V
43, 1breldm 4567 . . . . . . . 8  |-  ( ytpos 
F x  ->  y  e.  dom tpos  F )
5 dmtpos 5902 . . . . . . . . 9  |-  ( Rel 
dom  F  ->  dom tpos  F  =  `' dom  F )
65eleq2d 2123 . . . . . . . 8  |-  ( Rel 
dom  F  ->  ( y  e.  dom tpos  F  <->  y  e.  `' dom  F ) )
74, 6syl5ib 147 . . . . . . 7  |-  ( Rel 
dom  F  ->  ( ytpos 
F x  ->  y  e.  `' dom  F ) )
8 relcnv 4731 . . . . . . . 8  |-  Rel  `' dom  F
9 elrel 4470 . . . . . . . 8  |-  ( ( Rel  `' dom  F  /\  y  e.  `' dom  F )  ->  E. w E. z  y  =  <. w ,  z >.
)
108, 9mpan 408 . . . . . . 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 3795 . . . . . . . . 9  |-  ( y  =  <. w ,  z
>.  ->  ( ytpos  F x 
<-> 
<. w ,  z >.tpos  F x ) )
13 vex 2577 . . . . . . . . . 10  |-  w  e. 
_V
14 vex 2577 . . . . . . . . . 10  |-  z  e. 
_V
15 brtposg 5900 . . . . . . . . . 10  |-  ( ( w  e.  _V  /\  z  e.  _V  /\  x  e.  _V )  ->  ( <. w ,  z >.tpos  F x  <->  <. z ,  w >. F x ) )
1613, 14, 1, 15mp3an 1243 . . . . . . . . 9  |-  ( <.
w ,  z >.tpos  F x  <->  <. z ,  w >. F x )
1712, 16syl6bb 189 . . . . . . . 8  |-  ( y  =  <. w ,  z
>.  ->  ( ytpos  F x 
<-> 
<. z ,  w >. F x ) )
1814, 13opex 3994 . . . . . . . . 9  |-  <. z ,  w >.  e.  _V
1918, 1brelrn 4595 . . . . . . . 8  |-  ( <.
z ,  w >. F x  ->  x  e.  ran  F )
2017, 19syl6bi 156 . . . . . . 7  |-  ( y  =  <. w ,  z
>.  ->  ( ytpos  F x  ->  x  e.  ran  F ) )
2120exlimivv 1792 . . . . . 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 1716 . . . 4  |-  ( Rel 
dom  F  ->  ( E. y  ytpos  F x  ->  x  e.  ran  F ) )
242, 23syl5bi 145 . . 3  |-  ( Rel 
dom  F  ->  ( x  e.  ran tpos  F  ->  x  e.  ran  F ) )
251elrn 4605 . . . 4  |-  ( x  e.  ran  F  <->  E. y 
y F x )
263, 1breldm 4567 . . . . . . 7  |-  ( y F x  ->  y  e.  dom  F )
27 elrel 4470 . . . . . . . 8  |-  ( ( Rel  dom  F  /\  y  e.  dom  F )  ->  E. z E. w  y  =  <. z ,  w >. )
2827ex 112 . . . . . . 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 3795 . . . . . . . . 9  |-  ( y  =  <. z ,  w >.  ->  ( y F x  <->  <. z ,  w >. F x ) )
3130, 16syl6bbr 191 . . . . . . . 8  |-  ( y  =  <. z ,  w >.  ->  ( y F x  <->  <. w ,  z
>.tpos  F x ) )
3213, 14opex 3994 . . . . . . . . 9  |-  <. w ,  z >.  e.  _V
3332, 1brelrn 4595 . . . . . . . 8  |-  ( <.
w ,  z >.tpos  F x  ->  x  e. 
ran tpos  F )
3431, 33syl6bi 156 . . . . . . 7  |-  ( y  =  <. z ,  w >.  ->  ( y F x  ->  x  e.  ran tpos  F ) )
3534exlimivv 1792 . . . . . 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 1716 . . . 4  |-  ( Rel 
dom  F  ->  ( E. y  y F x  ->  x  e.  ran tpos  F ) )
3825, 37syl5bi 145 . . 3  |-  ( Rel 
dom  F  ->  ( x  e.  ran  F  ->  x  e.  ran tpos  F ) )
3924, 38impbid 124 . 2  |-  ( Rel 
dom  F  ->  ( x  e.  ran tpos  F  <->  x  e.  ran  F ) )
4039eqrdv 2054 1  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574   <.cop 3406   class class class wbr 3792   `'ccnv 4372   dom cdm 4373   ran crn 4374   Rel wrel 4378  tpos ctpos 5890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938  df-tpos 5891
This theorem is referenced by:  tposfo2  5913
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