Metamath Proof Explorer


Theorem fdmfisuppfi

Description: The support of a function with a finite domain is always finite. (Contributed by AV, 27-Apr-2019)

Ref Expression
Hypotheses fdmfisuppfi.f
|- ( ph -> F : D --> R )
fdmfisuppfi.d
|- ( ph -> D e. Fin )
fdmfisuppfi.z
|- ( ph -> Z e. V )
Assertion fdmfisuppfi
|- ( ph -> ( F supp Z ) e. Fin )

Proof

Step Hyp Ref Expression
1 fdmfisuppfi.f
 |-  ( ph -> F : D --> R )
2 fdmfisuppfi.d
 |-  ( ph -> D e. Fin )
3 fdmfisuppfi.z
 |-  ( ph -> Z e. V )
4 1 2 fexd
 |-  ( ph -> F e. _V )
5 suppimacnv
 |-  ( ( F e. _V /\ Z e. V ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
6 4 3 5 syl2anc
 |-  ( ph -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
7 2 1 fisuppfi
 |-  ( ph -> ( `' F " ( _V \ { Z } ) ) e. Fin )
8 6 7 eqeltrd
 |-  ( ph -> ( F supp Z ) e. Fin )