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Metamath Proof Explorer

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 ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 0 ..^ 𝑁 ) ⟶ 𝐵 ) → ( ♯ ‘ 𝐹 ) = 𝑁 )

Proof

Step Hyp Ref Expression
1 ffn ⊢ ( 𝐹 : ( 0 ..^ 𝑁 ) ⟶ 𝐵 → 𝐹 Fn ( 0 ..^ 𝑁 ) )
2 ffzo0hash ⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 Fn ( 0 ..^ 𝑁 ) ) → ( ♯ ‘ 𝐹 ) = 𝑁 )
3 1 2 sylan2 ⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 0 ..^ 𝑁 ) ⟶ 𝐵 ) → ( ♯ ‘ 𝐹 ) = 𝑁 )