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Theorem cnvopab 5099
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 5067 . 2  |-  Rel  `' { <. x ,  y
>.  |  ph }
2 relopab 4828 . 2  |-  Rel  { <. y ,  x >.  | 
ph }
3 opelopabsbOLD 4289 . . . 4  |-  ( <.
w ,  z >.  e.  { <. x ,  y
>.  |  ph }  <->  [ z  /  y ] [
w  /  x ] ph )
4 sbcom2 2066 . . . 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 2804 . . . 4  |-  z  e. 
_V
7 vex 2804 . . . 4  |-  w  e. 
_V
86, 7opelcnv 4879 . . 3  |-  ( <.
z ,  w >.  e.  `' { <. x ,  y
>.  |  ph }  <->  <. w ,  z >.  e.  { <. x ,  y >.  |  ph } )
9 opelopabsbOLD 4289 . . 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 4797 1  |-  `' { <. x ,  y >.  |  ph }  =  { <. y ,  x >.  | 
ph }
Colors of variables: wff set class
Syntax hints:    = wceq 1632   [wsb 1638    e. wcel 1696   <.cop 3656   {copab 4092   `'ccnv 4704
This theorem is referenced by:  cnvxp  5113  mptpreima  5182  f1ocnvd  6082  mapsncnv  6830  compsscnv  8013  fsumrev  12257  fsumshft  12258  pt1hmeo  17513  xkocnv  17521  lgsquadlem3  20611  cnvadj  22488  mptcnv  23043  f1o3d  23053  axcontlem2  24665  ranncnt  25386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713
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