Metamath Proof Explorer


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 eqtr3id
 |-  ( ( A e. V /\ B e. W ) -> `' { <. A , B >. } = { <. B , A >. } )