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 e. NN0 -> ( Ack ` M ) Fn NN0 )

Proof

Step	Hyp	Ref	Expression
1		ackendofnn0	|- ( M e. NN0 -> ( Ack ` M ) : NN0 --> NN0 )
2		ffn	|- ( ( Ack ` M ) : NN0 --> NN0 -> ( Ack ` M ) Fn NN0 )
3	1 2	syl	|- ( M e. NN0 -> ( Ack ` M ) Fn NN0 )