Theorem fnfzo0hashnn0

Description: The value of the size function on a half-open range of nonnegative integers is a nonnegative integer. (Contributed by AV, 10-Apr-2021)

	Ref	Expression
Assertion	fnfzo0hashnn0	|- ( F Fn ( 0 ..^ N ) -> ( # ` F ) e. NN0 )

Proof

Step	Hyp	Ref	Expression
1		hashfn	|- ( F Fn ( 0 ..^ N ) -> ( # ` F ) = ( # ` ( 0 ..^ N ) ) )
2		fzofi	|- ( 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 )