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Theorem rntpos 6997
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 3083 . . . . 5  |-  z  e. 
_V
21elrn 5094 . . . 4  |-  ( z  e.  ran tpos  F  <->  E. w  wtpos  F z )
3 vex 3083 . . . . . . . . 9  |-  w  e. 
_V
43, 1breldm 5058 . . . . . . . 8  |-  ( wtpos 
F z  ->  w  e.  dom tpos  F )
5 dmtpos 6996 . . . . . . . . 9  |-  ( Rel 
dom  F  ->  dom tpos  F  =  `' dom  F )
65eleq2d 2492 . . . . . . . 8  |-  ( Rel 
dom  F  ->  ( w  e.  dom tpos  F  <->  w  e.  `' dom  F ) )
74, 6syl5ib 222 . . . . . . 7  |-  ( Rel 
dom  F  ->  ( wtpos 
F z  ->  w  e.  `' dom  F ) )
8 relcnv 5226 . . . . . . . 8  |-  Rel  `' dom  F
9 elrel 4956 . . . . . . . 8  |-  ( ( Rel  `' dom  F  /\  w  e.  `' dom  F )  ->  E. x E. y  w  =  <. x ,  y >.
)
108, 9mpan 674 . . . . . . 7  |-  ( w  e.  `' dom  F  ->  E. x E. y  w  =  <. x ,  y >. )
117, 10syl6 34 . . . . . 6  |-  ( Rel 
dom  F  ->  ( wtpos 
F z  ->  E. x E. y  w  =  <. x ,  y >.
) )
12 breq1 4426 . . . . . . . . 9  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  <->  <. x ,  y
>.tpos  F z ) )
13 brtpos 6993 . . . . . . . . . 10  |-  ( z  e.  _V  ->  ( <. x ,  y >.tpos  F z  <->  <. y ,  x >. F z ) )
141, 13ax-mp 5 . . . . . . . . 9  |-  ( <.
x ,  y >.tpos  F z  <->  <. y ,  x >. F z )
1512, 14syl6bb 264 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  <->  <. y ,  x >. F z ) )
16 opex 4685 . . . . . . . . 9  |-  <. y ,  x >.  e.  _V
1716, 1brelrn 5084 . . . . . . . 8  |-  ( <.
y ,  x >. F z  ->  z  e.  ran  F )
1815, 17syl6bi 231 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  ( wtpos  F
z  ->  z  e.  ran  F ) )
1918exlimivv 1771 . . . . . 6  |-  ( E. x E. y  w  =  <. x ,  y
>.  ->  ( wtpos  F
z  ->  z  e.  ran  F ) )
2011, 19syli 38 . . . . 5  |-  ( Rel 
dom  F  ->  ( wtpos 
F z  ->  z  e.  ran  F ) )
2120exlimdv 1772 . . . 4  |-  ( Rel 
dom  F  ->  ( E. w  wtpos  F z  ->  z  e.  ran  F ) )
222, 21syl5bi 220 . . 3  |-  ( Rel 
dom  F  ->  ( z  e.  ran tpos  F  ->  z  e.  ran  F ) )
231elrn 5094 . . . 4  |-  ( z  e.  ran  F  <->  E. w  w F z )
243, 1breldm 5058 . . . . . . 7  |-  ( w F z  ->  w  e.  dom  F )
25 elrel 4956 . . . . . . . 8  |-  ( ( Rel  dom  F  /\  w  e.  dom  F )  ->  E. y E. x  w  =  <. y ,  x >. )
2625ex 435 . . . . . . 7  |-  ( Rel 
dom  F  ->  ( w  e.  dom  F  ->  E. y E. x  w  =  <. y ,  x >. ) )
2724, 26syl5 33 . . . . . 6  |-  ( Rel 
dom  F  ->  ( w F z  ->  E. y E. x  w  =  <. y ,  x >. ) )
28 breq1 4426 . . . . . . . . 9  |-  ( w  =  <. y ,  x >.  ->  ( w F z  <->  <. y ,  x >. F z ) )
2928, 14syl6bbr 266 . . . . . . . 8  |-  ( w  =  <. y ,  x >.  ->  ( w F z  <->  <. x ,  y
>.tpos  F z ) )
30 opex 4685 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
3130, 1brelrn 5084 . . . . . . . 8  |-  ( <.
x ,  y >.tpos  F z  ->  z  e.  ran tpos  F )
3229, 31syl6bi 231 . . . . . . 7  |-  ( w  =  <. y ,  x >.  ->  ( w F z  ->  z  e.  ran tpos  F ) )
3332exlimivv 1771 . . . . . 6  |-  ( E. y E. x  w  =  <. y ,  x >.  ->  ( w F z  ->  z  e.  ran tpos  F ) )
3427, 33syli 38 . . . . 5  |-  ( Rel 
dom  F  ->  ( w F z  ->  z  e.  ran tpos  F ) )
3534exlimdv 1772 . . . 4  |-  ( Rel 
dom  F  ->  ( E. w  w F z  ->  z  e.  ran tpos  F ) )
3623, 35syl5bi 220 . . 3  |-  ( Rel 
dom  F  ->  ( z  e.  ran  F  -> 
z  e.  ran tpos  F ) )
3722, 36impbid 193 . 2  |-  ( Rel 
dom  F  ->  ( z  e.  ran tpos  F  <->  z  e.  ran  F ) )
3837eqrdv 2419 1  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437   E.wex 1657    e. wcel 1872   _Vcvv 3080   <.cop 4004   class class class wbr 4423   `'ccnv 4852   dom cdm 4853   ran crn 4854   Rel wrel 4858  tpos ctpos 6983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-tpos 6984
This theorem is referenced by:  tposfo2  7007  oppchofcl  16144  oyoncl  16154
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