Theorem cnvopab 5145

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 5121	. 2 |- Rel `' { <. x , y >. | ph }
2		relopab 4862	. 2 |- Rel { <. y , x >. | ph }
3		opelopabsbALT 4359	. . . 4 |- ( <. w , z >. e. { <. x , y >. | ph } <-> [ z / y ] [ w / x ] ph )
4		sbcom2 2040	. . . 4 |- ( [ z / y ] [ w / x ] ph <-> [ w / x ] [ z / y ] ph )
5	3, 4	bitri 184	. . 3 |- ( <. w , z >. e. { <. x , y >. | ph } <-> [ w / x ] [ z / y ] ph )
6		vex 2806	. . . 4 |- z e. _V
7		vex 2806	. . . 4 |- w e. _V
8	6, 7	opelcnv 4918	. . 3 |- ( <. z , w >. e. `' { <. x , y >. | ph } <-> <. w , z >. e. { <. x , y >. | ph } )
9		opelopabsbALT 4359	. . 3 |- ( <. z , w >. e. { <. y , x >. | ph } <-> [ w / x ] [ z / y ] ph )
10	5, 8, 9	3bitr4i 212	. 2 |- ( <. z , w >. e. `' { <. x , y >. | ph } <-> <. z , w >. e. { <. y , x >. | ph } )
11	1, 2, 10	eqrelriiv 4826	1 |- `' { <. x , y >. | ph } = { <. y , x >. | ph }

Syntax hints:   = wceq 1398   [wsb 1810   e. wcel 2202   <.cop 3676   {copab 4154   `'ccnv 4730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739

This theorem is referenced by:  mptcnv  5146  cnvxp  5162  mptpreima  5237  f1ocnvd  6235  cnvoprab  6408  mapsncnv  6907  lgsquadlem3  15898