Metamath Proof Explorer


Theorem fvmap

Description: Function value for a member of a set exponentiation. (Contributed by Glauco Siliprandi, 21-Nov-2020)

Ref Expression
Hypotheses fvmap.a
|- ( ph -> A e. V )
fvmap.b
|- ( ph -> B e. W )
fvmap.f
|- ( ph -> F e. ( A ^m B ) )
fvmap.c
|- ( ph -> C e. B )
Assertion fvmap
|- ( ph -> ( F ` C ) e. A )

Proof

Step Hyp Ref Expression
1 fvmap.a
 |-  ( ph -> A e. V )
2 fvmap.b
 |-  ( ph -> B e. W )
3 fvmap.f
 |-  ( ph -> F e. ( A ^m B ) )
4 fvmap.c
 |-  ( ph -> C e. B )
5 id
 |-  ( ph -> ph )
6 elmapg
 |-  ( ( A e. V /\ B e. W ) -> ( F e. ( A ^m B ) <-> F : B --> A ) )
7 1 2 6 syl2anc
 |-  ( ph -> ( F e. ( A ^m B ) <-> F : B --> A ) )
8 3 7 mpbid
 |-  ( ph -> F : B --> A )
9 8 ffvelrnda
 |-  ( ( ph /\ C e. B ) -> ( F ` C ) e. A )
10 5 4 9 syl2anc
 |-  ( ph -> ( F ` C ) e. A )