Theorem fvpr2g

Description: The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017) (Proof shortened by BJ, 26-Sep-2024)

	Ref	Expression
Assertion	fvpr2g	|- ( ( B e. V /\ D e. W /\ A =/= B ) -> ( { <. A , C >. , <. B , D >. } ` B ) = D )

Proof

Step	Hyp	Ref	Expression
1		prcom	|- { <. A , C >. , <. B , D >. } = { <. B , D >. , <. A , C >. }
2	1	fveq1i	|- ( { <. A , C >. , <. B , D >. } ` B ) = ( { <. B , D >. , <. A , C >. } ` B )
3		necom	|- ( A =/= B <-> B =/= A )
4		fvpr1g	|- ( ( B e. V /\ D e. W /\ B =/= A ) -> ( { <. B , D >. , <. A , C >. } ` B ) = D )
5	3 4	syl3an3b	|- ( ( B e. V /\ D e. W /\ A =/= B ) -> ( { <. B , D >. , <. A , C >. } ` B ) = D )
6	2 5	eqtrid	|- ( ( B e. V /\ D e. W /\ A =/= B ) -> ( { <. A , C >. , <. B , D >. } ` B ) = D )