Theorem ffzo0hash

Description: The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018)

	Ref	Expression
Assertion	ffzo0hash	$$⊢\left(N\in {\mathbb{N}}_0\wedge FFn\left(0..^N\right)\right)\to \left|F\right|=N$$

Proof

Step	Hyp	Ref	Expression
1		hashfn	$$⊢FFn\left(0..^N\right)\to \left|F\right|=\left|\left(0..^N\right)\right|$$
2		hashfzo0	$$⊢N\in {\mathbb{N}}_0\to \left|\left(0..^N\right)\right|=N$$
3	1 2	sylan9eqr	$$⊢\left(N\in {\mathbb{N}}_0\wedge FFn\left(0..^N\right)\right)\to \left|F\right|=N$$