Equivalence of Definitions of Inverse Image Mapping of Relation

Proof
Consider the mapping defined as:
 * $\forall Y \in \powerset T: \map {\RR^\gets} Y = \set {s \in S: \exists t \in Y: \tuple {t, s} \in \RR^{-1} }$

Let $\Img \RR \cap Y = \O$.

Then:
 * $\forall t \in T: \neg \exists t \in \Img \RR \cap Y$

and so:
 * $\set {s \in S: \exists t \in Y: \tuple {s, t} \in \RR} = \O$

and so:
 * $\forall Y \in \powerset T: \map {\RR^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \tuple {t, s} \in \RR^{-1} } & : \Img f \cap Y \ne \O \\ \O & : \Img f \cap Y = \O \end {cases}$