Theorem fvtp1

Description: The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011)

	Ref	Expression
Hypotheses	fvtp1.1	|- A e. _V
	fvtp1.4	|- D e. _V
Assertion	fvtp1	|- ( ( A =/= B /\ A =/= C ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = D )

Proof

Step	Hyp	Ref	Expression
1		fvtp1.1	|- A e. _V
2		fvtp1.4	|- D e. _V
3		df-tp	|- { <. A , D >. , <. B , E >. , <. C , F >. } = ( { <. A , D >. , <. B , E >. } u. { <. C , F >. } )
4	3	fveq1i	|- ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = ( ( { <. A , D >. , <. B , E >. } u. { <. C , F >. } ) ` A )
5		necom	|- ( A =/= C <-> C =/= A )
6		fvunsn	|- ( C =/= A -> ( ( { <. A , D >. , <. B , E >. } u. { <. C , F >. } ) ` A ) = ( { <. A , D >. , <. B , E >. } ` A ) )
7	5 6	sylbi	|- ( A =/= C -> ( ( { <. A , D >. , <. B , E >. } u. { <. C , F >. } ) ` A ) = ( { <. A , D >. , <. B , E >. } ` A ) )
8	1 2	fvpr1	|- ( A =/= B -> ( { <. A , D >. , <. B , E >. } ` A ) = D )
9	7 8	sylan9eqr	|- ( ( A =/= B /\ A =/= C ) -> ( ( { <. A , D >. , <. B , E >. } u. { <. C , F >. } ) ` A ) = D )
10	4 9	eqtrid	|- ( ( A =/= B /\ A =/= C ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = D )