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 ) |
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 ) |