Theorem fvpr1

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	fvpr1.1	|- A e. _V
	fvpr1.2	|- C e. _V
Assertion	fvpr1	|- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` A ) = C )

Proof

Step	Hyp	Ref	Expression
1		fvpr1.1	|- A e. _V
2		fvpr1.2	|- C e. _V
3		fvpr1g	|- ( ( A e. _V /\ C e. _V /\ A =/= B ) -> ( { <. A , C >. , <. B , D >. } ` A ) = C )
4	1 2 3	mp3an12	|- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` A ) = C )