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Metamath Proof Explorer

Theorem ackfnnn0

Description: The Ackermann function at any nonnegative integer is a function on the nonnegative integers. (Contributed by AV, 4-May-2024) (Proof shortened by AV, 8-May-2024)

Ref Expression
Assertion ackfnnn0 ( 𝑀 ∈ ℕ0 → ( Ack ‘ 𝑀 ) Fn ℕ0 )

Proof

Step Hyp Ref Expression
1 ackendofnn0 ⊢ ( 𝑀 ∈ ℕ0 → ( Ack ‘ 𝑀 ) : ℕ0 ⟶ ℕ0 )
2 ffn ⊢ ( ( Ack ‘ 𝑀 ) : ℕ0 ⟶ ℕ0 → ( Ack ‘ 𝑀 ) Fn ℕ0 )
3 1 2 syl ⊢ ( 𝑀 ∈ ℕ0 → ( Ack ‘ 𝑀 ) Fn ℕ0 )