Theorem 1aryfvalel

Description: A unary (endo)function on a set X . (Contributed by AV, 15-May-2024)

	Ref	Expression
Assertion	1aryfvalel	$$⊢X\in V\to \left(F\in \left(1\mathrm{-aryF}X\right)\leftrightarrow F:X^{\left\{0\right\}}⟶X\right)$$

Proof

Step	Hyp	Ref	Expression
1		1nn0	$$⊢1\in {\mathbb{N}}_0$$
2		fzo01	$$⊢\left(0..^1\right)=\left\{0\right\}$$
3	2	eqcomi	$$⊢\left\{0\right\}=\left(0..^1\right)$$
4	3	naryfvalel	$$⊢\left(1\in {\mathbb{N}}_0\wedge X\in V\right)\to \left(F\in \left(1\mathrm{-aryF}X\right)\leftrightarrow F:X^{\left\{0\right\}}⟶X\right)$$
5	1 4	mpan	$$⊢X\in V\to \left(F\in \left(1\mathrm{-aryF}X\right)\leftrightarrow F:X^{\left\{0\right\}}⟶X\right)$$