Metamath Proof Explorer


Theorem mapfvd

Description: The value of a function that maps from B to A . (Contributed by AV, 2-Feb-2023)

Ref Expression
Hypotheses mapfvd.m
|- M = ( A ^m B )
mapfvd.f
|- ( ph -> F e. M )
mapfvd.x
|- ( ph -> X e. B )
Assertion mapfvd
|- ( ph -> ( F ` X ) e. A )

Proof

Step Hyp Ref Expression
1 mapfvd.m
 |-  M = ( A ^m B )
2 mapfvd.f
 |-  ( ph -> F e. M )
3 mapfvd.x
 |-  ( ph -> X e. B )
4 elmapi
 |-  ( F e. ( A ^m B ) -> F : B --> A )
5 ffvelrn
 |-  ( ( F : B --> A /\ X e. B ) -> ( F ` X ) e. A )
6 5 expcom
 |-  ( X e. B -> ( F : B --> A -> ( F ` X ) e. A ) )
7 3 4 6 syl2imc
 |-  ( F e. ( A ^m B ) -> ( ph -> ( F ` X ) e. A ) )
8 7 1 eleq2s
 |-  ( F e. M -> ( ph -> ( F ` X ) e. A ) )
9 2 8 mpcom
 |-  ( ph -> ( F ` X ) e. A )