Direct Image Mapping of Relation is Mapping

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

Let $$f_{\mathcal R}$$ be the mapping induced by $\mathcal R$:


 * $$f_{\mathcal R}: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right): f_{\mathcal R} \left({X}\right) = \mathcal R \left({X}\right)$$

Then $$f_{\mathcal R}$$ is indeed a mapping.

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

Let $$X \subseteq S$$, i.e. $$X \in \mathcal P \left({S}\right)$$.


 * Suppose $$X = \varnothing$$. Then $$\mathcal R \left({X}\right) = \varnothing \subseteq T$$, from Image of Null is Null.


 * Suppose $$X = S$$. Then $$\mathcal R \left({X}\right) = \operatorname{Im} \left({\mathcal R}\right) \subseteq T$$ from Image of Domain is Subset of Codomain.


 * Finally, suppose $$\varnothing \subset X \subset S$$. From Image is Subset of Codomain, again we see that $$\mathcal R \left({X}\right) \subseteq T$$.

Now, from the definition of the power set, we have that $$Y \subseteq T \iff Y \in \mathcal P \left({T}\right)$$.

We defined $$f_\mathcal R \subseteq \mathcal P \left({S}\right) \times \mathcal P \left({T}\right)$$ such that:


 * $$f_{\mathcal R} : \mathcal P \left({S}\right) \to \mathcal P \left({T}\right): f_{\mathcal R} \left({X}\right) = \mathcal R \left({X}\right)$$

Clearly, by definition, there is only one $$\mathcal R \left({X}\right)$$ for any given $$X$$, and so $$f_{\mathcal R}$$ is functional.

We have shown that $$\forall X \subseteq S: \mathcal R \left({X}\right) \in \mathcal P \left({T}\right)$$.

So:
 * $$\forall X \in \mathcal P \left({S}\right): \exists_1 Y \in \mathcal P \left({T}\right): \mathcal R \left({X}\right) = Y$$

and thus:
 * $$\forall X \in \mathcal P \left({S}\right): \exists_1 Y \in \mathcal P \left({T}\right): f_{\mathcal R} \left({X}\right) = Y$$.

So:
 * $$f_{\mathcal R}$$ is defined for all $$X \in \mathcal P \left({S}\right)$$

and therefore $$f_{\mathcal R}$$ is a mapping.

Also see

 * Mapping Induced on Power Set by Inverse Relation