Description: The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | 0fsupp | |- ( Z e. V -> (/) finSupp Z ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supp0 | |- ( Z e. V -> ( (/) supp Z ) = (/) ) |
|
2 | 0fin | |- (/) e. Fin |
|
3 | 1 2 | eqeltrdi | |- ( Z e. V -> ( (/) supp Z ) e. Fin ) |
4 | fun0 | |- Fun (/) |
|
5 | 0ex | |- (/) e. _V |
|
6 | funisfsupp | |- ( ( Fun (/) /\ (/) e. _V /\ Z e. V ) -> ( (/) finSupp Z <-> ( (/) supp Z ) e. Fin ) ) |
|
7 | 4 5 6 | mp3an12 | |- ( Z e. V -> ( (/) finSupp Z <-> ( (/) supp Z ) e. Fin ) ) |
8 | 3 7 | mpbird | |- ( Z e. V -> (/) finSupp Z ) |