Theorem 2aryfvalel

Description: A binary (endo)function on a set X . (Contributed by AV, 20-May-2024)

	Ref	Expression
Assertion	2aryfvalel	$$⊢X\in V\to \left(F\in \left(2\mathrm{-aryF}X\right)\leftrightarrow F:X^{\left\{0,1\right\}}⟶X\right)$$

Proof

Step	Hyp	Ref	Expression
1		2nn0	$$⊢2\in {\mathbb{N}}_0$$
2		fzo0to2pr	$$⊢\left(0..^2\right)=\left\{0,1\right\}$$
3	2	eqcomi	$$⊢\left\{0,1\right\}=\left(0..^2\right)$$
4	3	naryfvalel	$$⊢\left(2\in {\mathbb{N}}_0\wedge X\in V\right)\to \left(F\in \left(2\mathrm{-aryF}X\right)\leftrightarrow F:X^{\left\{0,1\right\}}⟶X\right)$$
5	1 4	mpan	$$⊢X\in V\to \left(F\in \left(2\mathrm{-aryF}X\right)\leftrightarrow F:X^{\left\{0,1\right\}}⟶X\right)$$