MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvopab Structured version   Unicode version

Theorem cnvopab 5248
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 5218 . 2  |-  Rel  `' { <. x ,  y
>.  |  ph }
2 relopab 4971 . 2  |-  Rel  { <. y ,  x >.  | 
ph }
3 opelopabsbALT 4721 . . . 4  |-  ( <.
w ,  z >.  e.  { <. x ,  y
>.  |  ph }  <->  [ z  /  y ] [
w  /  x ] ph )
4 sbcom2 2238 . . . 4  |-  ( [ z  /  y ] [ w  /  x ] ph  <->  [ w  /  x ] [ z  /  y ] ph )
53, 4bitri 252 . . 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 5027 . . 3  |-  ( <.
z ,  w >.  e.  `' { <. x ,  y
>.  |  ph }  <->  <. w ,  z >.  e.  { <. x ,  y >.  |  ph } )
9 opelopabsbALT 4721 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. y ,  x >.  |  ph }  <->  [ w  /  x ] [ z  /  y ] ph )
105, 8, 93bitr4i 280 . 2  |-  ( <.
z ,  w >.  e.  `' { <. x ,  y
>.  |  ph }  <->  <. z ,  w >.  e.  { <. y ,  x >.  |  ph } )
111, 2, 10eqrelriiv 4940 1  |-  `' { <. x ,  y >.  |  ph }  =  { <. y ,  x >.  | 
ph }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   [wsb 1786    e. wcel 1867   <.cop 3999   {copab 4474   `'ccnv 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-rel 4852  df-cnv 4853
This theorem is referenced by:  mptcnv  5249  cnvxp  5265  mptpreima  5339  f1ocnvd  6523  mapsncnv  7517  compsscnv  8790  dfiso2  15629  xkocnv  20766  lgsquadlem3  24186  axcontlem2  24882  cnvadj  27421  f1o3d  28110  cnvoprab  28192
  Copyright terms: Public domain W3C validator