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Theorem cnvopab 5407
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 5374 . 2  |-  Rel  `' { <. x ,  y
>.  |  ph }
2 relopab 5129 . 2  |-  Rel  { <. y ,  x >.  | 
ph }
3 opelopabsbALT 4756 . . . 4  |-  ( <.
w ,  z >.  e.  { <. x ,  y
>.  |  ph }  <->  [ z  /  y ] [
w  /  x ] ph )
4 sbcom2 2173 . . . 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 3116 . . . 4  |-  z  e. 
_V
7 vex 3116 . . . 4  |-  w  e. 
_V
86, 7opelcnv 5184 . . 3  |-  ( <.
z ,  w >.  e.  `' { <. x ,  y
>.  |  ph }  <->  <. w ,  z >.  e.  { <. x ,  y >.  |  ph } )
9 opelopabsbALT 4756 . . 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 5097 1  |-  `' { <. x ,  y >.  |  ph }  =  { <. y ,  x >.  | 
ph }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   [wsb 1711    e. wcel 1767   <.cop 4033   {copab 4504   `'ccnv 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007
This theorem is referenced by:  mptcnv  5408  cnvxp  5424  mptpreima  5500  f1ocnvd  6509  mapsncnv  7466  compsscnv  8752  fsumrev  13560  pt1hmeo  20134  xkocnv  20142  lgsquadlem3  23456  axcontlem2  24041  cnvadj  26584  f1o3d  27241  cnvoprab  27315  fprodshft  28959  fprodrev  28960
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