Theorem cnvsng

Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015) (Proof shortened by BJ, 12-Feb-2022)

	Ref	Expression
Assertion	cnvsng	|- ( ( A e. V /\ B e. W ) -> `' { <. A , B >. } = { <. B , A >. } )

Proof

Step	Hyp	Ref	Expression
1		cnvcnvsn	|- `' `' { <. B , A >. } = `' { <. A , B >. }
2		relsnopg	|- ( ( B e. W /\ A e. V ) -> Rel { <. B , A >. } )
3	2	ancoms	|- ( ( A e. V /\ B e. W ) -> Rel { <. B , A >. } )
4		dfrel2	|- ( Rel { <. B , A >. } <-> `' `' { <. B , A >. } = { <. B , A >. } )
5	3 4	sylib	|- ( ( A e. V /\ B e. W ) -> `' `' { <. B , A >. } = { <. B , A >. } )
6	1 5	syl5eqr	|- ( ( A e. V /\ B e. W ) -> `' { <. A , B >. } = { <. B , A >. } )