Surjection Induced by Powerset is Induced by Surjection

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

Let $\mathcal R^\to: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ be the mapping induced on $\mathcal P \left({S}\right)$ by $\mathcal R$.

Let $\mathcal R^\to$ be a surjection.

Let $X = \operatorname{Im}^{-1} \left({\mathcal R}\right)$, that is, the preimage of $\mathcal R$.

Then $\mathcal R {\restriction_X} \subseteq X \times T$, that is, the restriction of $\mathcal R$ to $X$, is a surjection.

Proof
Let $X$ be the preimage of $\mathcal R$.

Suppose $\mathcal R {\restriction_X} \subseteq X \times T$ is a mapping, but not a surjection.

Then:
 * $\exists y \in T: \neg \exists x \in S: \left({x, y}\right) \in \mathcal R$

Because no element of $S$ relates to $y$, no subset of $S$ contains any element of $S$ that relates to the subset $\left\{{y}\right\} \subseteq T$.

Thus:
 * $\exists \left\{{y}\right\} \in \mathcal P \left({T}\right): \neg \exists X \in \mathcal P \left({S}\right): \mathcal R^\to \left({X}\right) = \left\{{y}\right\}$

So we see that $\mathcal R^\to: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ is not a surjection.

Now, suppose $\mathcal R {\restriction_X} \subseteq X \times T$ is not even a mapping.

This could happen, according to the definition of a mapping, in one of two ways:


 * $(1): \quad \exists x \in X: \neg \exists y \in T: \left({x, y}\right) \in \mathcal R$
 * $(2): \quad \exists x \in S: \left({x, y_1}\right) \in \mathcal R \land \left({x, y_2}\right) \in \mathcal R: y_1 \ne y_2$

Because $X$ is already the preimage of $\mathcal R$, the first of these cannot happen here.

For the second, it can be seen that neither $\left\{{y_1}\right\}$ nor $\left\{{y_2}\right\}$ can be in $\operatorname{Rng} \left({\mathcal R^\to \left({\mathcal P \left({S}\right)}\right)}\right)$.

Therefore $\mathcal R^\to: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ can not be a surjection.

Thus, by the Rule of Transposition, the result follows.