Theorem naryrcl

Description: Reverse closure for n-ary (endo)functions. (Contributed by AV, 14-May-2024)

	Ref	Expression
Hypothesis	naryfval.i	$$⊢I=\left(0..^N\right)$$
Assertion	naryrcl	$$⊢F\in \left(N\mathrm{-aryF}X\right)\to \left(N\in {\mathbb{N}}_0\wedge X\in \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		naryfval.i	$$⊢I=\left(0..^N\right)$$
2		df-naryf	$$⊢\mathrm{-aryF}=\left(x\in {\mathbb{N}}_0,n\in \mathrm{V}⟼n^{\left(n^{\left(0..^x\right)}\right)}\right)$$
3	2	elmpocl	$$⊢F\in \left(N\mathrm{-aryF}X\right)\to \left(N\in {\mathbb{N}}_0\wedge X\in \mathrm{V}\right)$$