Theorem fvpr2

Description: The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010)

	Ref	Expression
Hypotheses	fvpr2.1	|- B e. _V
	fvpr2.2	|- D e. _V
Assertion	fvpr2	|- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` B ) = D )

Proof

Step	Hyp	Ref	Expression
1		fvpr2.1	|- B e. _V
2		fvpr2.2	|- D e. _V
3		prcom	|- { <. A , C >. , <. B , D >. } = { <. B , D >. , <. A , C >. }
4	3	fveq1i	|- ( { <. A , C >. , <. B , D >. } ` B ) = ( { <. B , D >. , <. A , C >. } ` B )
5		necom	|- ( A =/= B <-> B =/= A )
6	1 2	fvpr1	|- ( B =/= A -> ( { <. B , D >. , <. A , C >. } ` B ) = D )
7	5 6	sylbi	|- ( A =/= B -> ( { <. B , D >. , <. A , C >. } ` B ) = D )
8	4 7	eqtrid	|- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` B ) = D )