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 )