Theorem ffz0hash

Description: The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018) (Proof shortened by AV, 11-Apr-2021)

	Ref	Expression
Assertion	ffz0hash	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 0 ... 𝑁 ) ⟶ 𝐵 ) → ( ♯ ‘ 𝐹 ) = ( 𝑁 + 1 ) )

Proof

Step	Hyp	Ref	Expression
1		ffn	⊢ ( 𝐹 : ( 0 ... 𝑁 ) ⟶ 𝐵 → 𝐹 Fn ( 0 ... 𝑁 ) )
2		fnfz0hash	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 Fn ( 0 ... 𝑁 ) ) → ( ♯ ‘ 𝐹 ) = ( 𝑁 + 1 ) )
3	1 2	sylan2	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 0 ... 𝑁 ) ⟶ 𝐵 ) → ( ♯ ‘ 𝐹 ) = ( 𝑁 + 1 ) )