Direct Image Mapping of Relation is Mapping

Theorem
Let $S$ and $T$ be sets.

Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$.

Let $\mathcal R^\to: \powerset S \to \powerset T$ be the direct image mapping of $\mathcal R$:


 * $\forall X \in \powerset S: \map {\mathcal R^\to} X = \set {t \in T: \exists s \in X: \tuple {s, t} \in \mathcal R}$

Then $\mathcal R^\to$ is indeed a mapping.

Proof
Take the general relation $\mathcal R \subseteq S \times T$.

Let $X \subseteq S$, that is $X \in \powerset S$.

Suppose $X = \O$.

Then from Image of Empty Set is Empty Set:
 * $\map {\mathcal R^\to} X = \O \subseteq T$

Suppose $X = S$.

Then from Image is Subset of Codomain: Corollary 1:
 * $\map {\mathcal R^\to} X = \Img {\mathcal R} \subseteq T$

Finally, suppose $\O \subset X \subset S$.

From Image is Subset of Codomain, again:
 * $\map {\mathcal R^\to} X \subseteq T$

Now, from the definition of the power set, we have that:
 * $Y \subseteq T \iff Y \in \powerset T$

We defined $R^\to \subseteq \powerset S \times \powerset T$ such that:


 * $R^\to: \powerset S \to \powerset T: \map {\mathcal R^\to} X = \set {t \in T: \exists s \in X: \tuple {s, t} \in \mathcal R}$

By definition, there is only one $\map {\mathcal R^\to} X$ for any given $X$, and so $R^\to$ is many-to-one.

We have shown that:
 * $\forall X \subseteq S: \map {\mathcal R^\to} X \in \powerset T$

So:
 * $\forall X \in \powerset S: \exists_1 Y \in \powerset T: \map {\mathcal R^\to} X = Y$

So:
 * $R^\to$ is defined for all $X \in \powerset S$

and therefore $R^\to$ is a mapping.

Also see

 * Inverse Image Mapping of Relation is Mapping