Theorem cnvopab 5138

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 5114	. 2 |- Rel `' { <. x , y >. | ph }
2		relopab 4856	. 2 |- Rel { <. y , x >. | ph }
3		opelopabsbALT 4353	. . . 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 2805	. . . 4 |- z e. _V
7		vex 2805	. . . 4 |- w e. _V
8	6, 7	opelcnv 4912	. . 3 |- ( <. z , w >. e. `' { <. x , y >. | ph } <-> <. w , z >. e. { <. x , y >. | ph } )
9		opelopabsbALT 4353	. . 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 4820	1 |- `' { <. x , y >. | ph } = { <. y , x >. | ph }

Syntax hints:   = wceq 1397   [wsb 1810   e. wcel 2202   <.cop 3672   {copab 4149   `'ccnv 4724

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733

This theorem is referenced by:  mptcnv  5139  cnvxp  5155  mptpreima  5230  f1ocnvd  6224  cnvoprab  6398  mapsncnv  6863  lgsquadlem3  15807