Theorem cnvopab

Description: The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	cnvopab	|- `' { <. x , y >. | ph } = { <. y , x >. | ph }

Proof

Step	Hyp	Ref	Expression
1		relcnv	|- Rel `' { <. x , y >. | ph }
2		relopabv	|- Rel { <. y , x >. | ph }
3		vopelopabsb	|- ( <. w , z >. e. { <. x , y >. | ph } <-> [ w / x ] [ z / y ] ph )
4		sbcom2	|- ( [ w / x ] [ z / y ] ph <-> [ z / y ] [ w / x ] ph )
5	3 4	bitri	|- ( <. w , z >. e. { <. x , y >. | ph } <-> [ z / y ] [ w / x ] ph )
6		vex	|- z e. _V
7		vex	|- w e. _V
8	6 7	opelcnv	|- ( <. z , w >. e. `' { <. x , y >. | ph } <-> <. w , z >. e. { <. x , y >. | ph } )
9		vopelopabsb	|- ( <. z , w >. e. { <. y , x >. | ph } <-> [ z / y ] [ w / x ] ph )
10	5 8 9	3bitr4i	|- ( <. z , w >. e. `' { <. x , y >. | ph } <-> <. z , w >. e. { <. y , x >. | ph } )
11	1 2 10	eqrelriiv	|- `' { <. x , y >. | ph } = { <. y , x >. | ph }