Metamath Proof Explorer

Theorem fnfz0hashnn0

Description: The size of a function on a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by AV, 10-Apr-2021)

Ref Expression
Assertion fnfz0hashnn0 
|- ( F Fn ( 0 ... N ) -> ( # ` F ) e. NN0 )

Proof

Step Hyp Ref Expression
1 hashfn 
 |-  ( F Fn ( 0 ... N ) -> ( # ` F ) = ( # ` ( 0 ... N ) ) )
2 fzfi 
 |-  ( 0 ... N ) e. Fin
3 hashcl 
 |-  ( ( 0 ... N ) e. Fin -> ( # ` ( 0 ... N ) ) e. NN0 )
4 2 3 ax-mp 
 |-  ( # ` ( 0 ... N ) ) e. NN0
5 1 4 eqeltrdi 
 |-  ( F Fn ( 0 ... N ) -> ( # ` F ) e. NN0 )