Theorem cnvopab 5533

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.

Step	Hyp	Ref	Expression
1		relcnv 5503	. 2 |- Rel `' { <. x , y >. | ph }
2		relopab 5247	. 2 |- Rel { <. y , x >. | ph }
3		opelopabsbALT 4984	. . . 4 |- ( <. w , z >. e. { <. x , y >. | ph } <-> [ z / y ] [ w / x ] ph )
4		sbcom2 2445	. . . 4 |- ( [ z / y ] [ w / x ] ph <-> [ w / x ] [ z / y ] ph )
5	3, 4	bitri 264	. . 3 |- ( <. w , z >. e. { <. x , y >. | ph } <-> [ w / x ] [ z / y ] ph )
6		vex 3203	. . . 4 |- z e. _V
7		vex 3203	. . . 4 |- w e. _V
8	6, 7	opelcnv 5304	. . 3 |- ( <. z , w >. e. `' { <. x , y >. | ph } <-> <. w , z >. e. { <. x , y >. | ph } )
9		opelopabsbALT 4984	. . 3 |- ( <. z , w >. e. { <. y , x >. | ph } <-> [ w / x ] [ z / y ] ph )
10	5, 8, 9	3bitr4i 292	. 2 |- ( <. z , w >. e. `' { <. x , y >. | ph } <-> <. z , w >. e. { <. y , x >. | ph } )
11	1, 2, 10	eqrelriiv 5214	1 |- `' { <. x , y >. | ph } = { <. y , x >. | ph }

Syntax hints:   = wceq 1483   [wsb 1880   e. wcel 1990   <.cop 4183   {copab 4712   `'ccnv 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by:  mptcnv  5534  cnvxp  5551  mptpreima  5628  f1ocnvd  6884  mapsncnv  7904  compsscnv  9193  dfiso2  16432  xkocnv  21617  lgsquadlem3  25107  axcontlem2  25845  cnvadj  28751  f1o3d  29431  cnvoprab  29498  fsovrfovd  38303