Unit of Ring of Mappings iff Image is Subset of Ring Units

Theorem
Let $\struct {R, +, \circ}$ be a ring with unity $1_R$.

Let $U_R$ be the set of units in $R$.

Let $S$ be a set.

Let $\struct {R^S, +', \circ'}$ be the ring of mappings on the set of mappings $R^S$.

Then:
 * $f \in R^S$ is a unit of $R^S$ $\Img f \subseteq U_R$

where $\Img f$ is the image of $f$.

In which case, the inverse of $f$ is the mapping $f^{-1} : S \to U_R$ defined by:
 * $\forall x \in S : \map {f^{-1}} {x} = \map f x^{-1}$

Proof
From Structure Induced by Ring with Unity Operations is Ring with Unity, $\struct {R^S, +', \circ'}$ has a unity $f_{1_R}$ defined by:
 * $\forall x \in S: \map {f_{1_R}} x = 1_R$

=== Unit of Ring of Mappings implies Image is Subset of Ring Units ===

=== Image is Subset of Ring Units implies Unit of Ring of Mappings ===