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Theorem cnvopab 5347
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 5315 . 2  |-  Rel  `' { <. x ,  y
>.  |  ph }
2 relopab 5075 . 2  |-  Rel  { <. y ,  x >.  | 
ph }
3 opelopabsbALT 4707 . . . 4  |-  ( <.
w ,  z >.  e.  { <. x ,  y
>.  |  ph }  <->  [ z  /  y ] [
w  /  x ] ph )
4 sbcom2 2160 . . . 4  |-  ( [ z  /  y ] [ w  /  x ] ph  <->  [ w  /  x ] [ z  /  y ] ph )
53, 4bitri 249 . . 3  |-  ( <.
w ,  z >.  e.  { <. x ,  y
>.  |  ph }  <->  [ w  /  x ] [ z  /  y ] ph )
6 vex 3081 . . . 4  |-  z  e. 
_V
7 vex 3081 . . . 4  |-  w  e. 
_V
86, 7opelcnv 5130 . . 3  |-  ( <.
z ,  w >.  e.  `' { <. x ,  y
>.  |  ph }  <->  <. w ,  z >.  e.  { <. x ,  y >.  |  ph } )
9 opelopabsbALT 4707 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. y ,  x >.  |  ph }  <->  [ w  /  x ] [ z  /  y ] ph )
105, 8, 93bitr4i 277 . 2  |-  ( <.
z ,  w >.  e.  `' { <. x ,  y
>.  |  ph }  <->  <. z ,  w >.  e.  { <. y ,  x >.  |  ph } )
111, 2, 10eqrelriiv 5043 1  |-  `' { <. x ,  y >.  |  ph }  =  { <. y ,  x >.  | 
ph }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   [wsb 1702    e. wcel 1758   <.cop 3992   {copab 4458   `'ccnv 4948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-cnv 4957
This theorem is referenced by:  mptcnv  5348  cnvxp  5364  mptpreima  5440  f1ocnvd  6420  mapsncnv  7370  compsscnv  8652  fsumrev  13365  pt1hmeo  19512  xkocnv  19520  lgsquadlem3  22829  axcontlem2  23364  cnvadj  25449  f1o3d  26100  cnvoprab  26175  fprodshft  27632  fprodrev  27633
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