Metamath Proof Explorer

Theorem fvtp1g

Description: The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017)

Ref Expression
Assertion fvtp1g 
|- ( ( ( A e. V /\ D e. W ) /\ ( A =/= B /\ A =/= C ) ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = D )

Proof

Step Hyp Ref Expression
1 df-tp 
 |-  { <. A , D >. , <. B , E >. , <. C , F >. } = ( { <. A , D >. , <. B , E >. } u. { <. C , F >. } )
2 1 fveq1i 
 |-  ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = ( ( { <. A , D >. , <. B , E >. } u. { <. C , F >. } ) ` A )
3 necom 
 |-  ( A =/= C <-> C =/= A )
4 fvunsn 
 |-  ( C =/= A -> ( ( { <. A , D >. , <. B , E >. } u. { <. C , F >. } ) ` A ) = ( { <. A , D >. , <. B , E >. } ` A ) )
5 3 4 sylbi 
 |-  ( A =/= C -> ( ( { <. A , D >. , <. B , E >. } u. { <. C , F >. } ) ` A ) = ( { <. A , D >. , <. B , E >. } ` A ) )
6 5 ad2antll 
 |-  ( ( ( A e. V /\ D e. W ) /\ ( A =/= B /\ A =/= C ) ) -> ( ( { <. A , D >. , <. B , E >. } u. { <. C , F >. } ) ` A ) = ( { <. A , D >. , <. B , E >. } ` A ) )
7 fvpr1g 
 |-  ( ( A e. V /\ D e. W /\ A =/= B ) -> ( { <. A , D >. , <. B , E >. } ` A ) = D )
8 7 3expa 
 |-  ( ( ( A e. V /\ D e. W ) /\ A =/= B ) -> ( { <. A , D >. , <. B , E >. } ` A ) = D )
9 8 adantrr 
 |-  ( ( ( A e. V /\ D e. W ) /\ ( A =/= B /\ A =/= C ) ) -> ( { <. A , D >. , <. B , E >. } ` A ) = D )
10 6 9 eqtrd 
 |-  ( ( ( A e. V /\ D e. W ) /\ ( A =/= B /\ A =/= C ) ) -> ( ( { <. A , D >. , <. B , E >. } u. { <. C , F >. } ) ` A ) = D )
11 2 10 eqtrid 
 |-  ( ( ( A e. V /\ D e. W ) /\ ( A =/= B /\ A =/= C ) ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = D )