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Theorem cnvopab 5083
Description: The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvopab  |-  `' { <. x ,  y >.  |  ph }  =  { <. y ,  x >.  | 
ph }
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem cnvopab
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5051 . 2  |-  Rel  `' { <. x ,  y
>.  |  ph }
2 relopab 4812 . 2  |-  Rel  { <. y ,  x >.  | 
ph }
3 opelopabsbOLD 4273 . . . 4  |-  ( <.
w ,  z >.  e.  { <. x ,  y
>.  |  ph }  <->  [ z  /  y ] [
w  /  x ] ph )
4 sbcom2 2053 . . . 4  |-  ( [ z  /  y ] [ w  /  x ] ph  <->  [ w  /  x ] [ z  /  y ] ph )
53, 4bitri 240 . . 3  |-  ( <.
w ,  z >.  e.  { <. x ,  y
>.  |  ph }  <->  [ w  /  x ] [ z  /  y ] ph )
6 vex 2791 . . . 4  |-  z  e. 
_V
7 vex 2791 . . . 4  |-  w  e. 
_V
86, 7opelcnv 4863 . . 3  |-  ( <.
z ,  w >.  e.  `' { <. x ,  y
>.  |  ph }  <->  <. w ,  z >.  e.  { <. x ,  y >.  |  ph } )
9 opelopabsbOLD 4273 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. y ,  x >.  |  ph }  <->  [ w  /  x ] [ z  /  y ] ph )
105, 8, 93bitr4i 268 . 2  |-  ( <.
z ,  w >.  e.  `' { <. x ,  y
>.  |  ph }  <->  <. z ,  w >.  e.  { <. y ,  x >.  |  ph } )
111, 2, 10eqrelriiv 4781 1  |-  `' { <. x ,  y >.  |  ph }  =  { <. y ,  x >.  | 
ph }
Colors of variables: wff set class
Syntax hints:    = wceq 1623   [wsb 1629    e. wcel 1684   <.cop 3643   {copab 4076   `'ccnv 4688
This theorem is referenced by:  cnvxp  5097  mptpreima  5166  f1ocnvd  6066  mapsncnv  6814  compsscnv  7997  fsumrev  12241  fsumshft  12242  pt1hmeo  17497  xkocnv  17505  lgsquadlem3  20595  cnvadj  22472  mptcnv  23027  f1o3d  23037  axcontlem2  24593  ranncnt  25283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697
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