Metamath Proof Explorer

Theorem ffi

Description: A function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion ffi 
|- ( ( F : A --> B /\ A e. Fin ) -> F e. Fin )

Proof

Step Hyp Ref Expression
1 ffn 
 |-  ( F : A --> B -> F Fn A )
2 fnfi 
 |-  ( ( F Fn A /\ A e. Fin ) -> F e. Fin )
3 1 2 sylan 
 |-  ( ( F : A --> B /\ A e. Fin ) -> F e. Fin )