Equivalence of Definitions of Inverse Image Mapping of Mapping

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

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

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

and so:
 * $\set {s \in S: \exists t \in Y: \map f s = t} = \O$

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