Theorem fnfz0hash

Description: The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018)

	Ref	Expression
Assertion	fnfz0hash	$$⊢\left(N\in {\mathbb{N}}_0\wedge FFn\left(0\dots N\right)\right)\to \left|F\right|=N+1$$

Proof

Step	Hyp	Ref	Expression
1		hashfn	$$⊢FFn\left(0\dots N\right)\to \left|F\right|=\left|\left(0\dots N\right)\right|$$
2		hashfz0	$$⊢N\in {\mathbb{N}}_0\to \left|\left(0\dots N\right)\right|=N+1$$
3	1 2	sylan9eqr	$$⊢\left(N\in {\mathbb{N}}_0\wedge FFn\left(0\dots N\right)\right)\to \left|F\right|=N+1$$