Theorem fvpr2

Description: The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010) (Proof shortened by BJ, 26-Sep-2024)

	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		fvpr2g	|- ( ( B e. _V /\ D e. _V /\ A =/= B ) -> ( { <. A , C >. , <. B , D >. } ` B ) = D )
4	1 2 3	mp3an12	|- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` B ) = D )