Theorem fnfzo0hash

Description: The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018) (Proof shortened by AV, 11-Apr-2021)

	Ref	Expression
Assertion	fnfzo0hash	$$⊢\left(N\in {\mathbb{N}}_0\wedge F:\left(0..^N\right)⟶B\right)\to \left|F\right|=N$$

Proof

Step	Hyp	Ref	Expression
1		ffn	$$⊢F:\left(0..^N\right)⟶B\to FFn\left(0..^N\right)$$
2		ffzo0hash	$$⊢\left(N\in {\mathbb{N}}_0\wedge FFn\left(0..^N\right)\right)\to \left|F\right|=N$$
3	1 2	sylan2	$$⊢\left(N\in {\mathbb{N}}_0\wedge F:\left(0..^N\right)⟶B\right)\to \left|F\right|=N$$