Metamath Proof Explorer

Theorem fvtp2g

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 fvtp2g 
|- ( ( ( B e. V /\ E e. W ) /\ ( A =/= B /\ B =/= C ) ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` B ) = E )

Proof

Step Hyp Ref Expression
1 tprot 
 |-  { <. A , D >. , <. B , E >. , <. C , F >. } = { <. B , E >. , <. C , F >. , <. A , D >. }
2 1 fveq1i 
 |-  ( { <. A , D >. , <. B , E >. , <. C , F >. } ` B ) = ( { <. B , E >. , <. C , F >. , <. A , D >. } ` B )
3 necom 
 |-  ( A =/= B <-> B =/= A )
4 fvtp1g 
 |-  ( ( ( B e. V /\ E e. W ) /\ ( B =/= C /\ B =/= A ) ) -> ( { <. B , E >. , <. C , F >. , <. A , D >. } ` B ) = E )
5 4 expcom 
 |-  ( ( B =/= C /\ B =/= A ) -> ( ( B e. V /\ E e. W ) -> ( { <. B , E >. , <. C , F >. , <. A , D >. } ` B ) = E ) )
6 5 ancoms 
 |-  ( ( B =/= A /\ B =/= C ) -> ( ( B e. V /\ E e. W ) -> ( { <. B , E >. , <. C , F >. , <. A , D >. } ` B ) = E ) )
7 3 6 sylanb 
 |-  ( ( A =/= B /\ B =/= C ) -> ( ( B e. V /\ E e. W ) -> ( { <. B , E >. , <. C , F >. , <. A , D >. } ` B ) = E ) )
8 7 impcom 
 |-  ( ( ( B e. V /\ E e. W ) /\ ( A =/= B /\ B =/= C ) ) -> ( { <. B , E >. , <. C , F >. , <. A , D >. } ` B ) = E )
9 2 8 eqtrid 
 |-  ( ( ( B e. V /\ E e. W ) /\ ( A =/= B /\ B =/= C ) ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` B ) = E )