Theorem fvpr1

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

	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		df-pr	|- { <. A , C >. , <. B , D >. } = ( { <. A , C >. } u. { <. B , D >. } )
4	3	fveq1i	|- ( { <. A , C >. , <. B , D >. } ` A ) = ( ( { <. A , C >. } u. { <. B , D >. } ) ` A )
5		necom	|- ( A =/= B <-> B =/= A )
6		fvunsn	|- ( B =/= A -> ( ( { <. A , C >. } u. { <. B , D >. } ) ` A ) = ( { <. A , C >. } ` A ) )
7	5 6	sylbi	|- ( A =/= B -> ( ( { <. A , C >. } u. { <. B , D >. } ) ` A ) = ( { <. A , C >. } ` A ) )
8	4 7	syl5eq	|- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` A ) = ( { <. A , C >. } ` A ) )
9	1 2	fvsn	|- ( { <. A , C >. } ` A ) = C
10	8 9	eqtrdi	|- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` A ) = C )