Restriction of Mapping is Mapping

Theorem
Let $f: S \to T$ be a mapping.

Let $X \subseteq S$.

Let $f \restriction_X$ be the restriction of $f$ to $X$.

Then $f \restriction_X: X \to T$ is a mapping:
 * whose domain is $X$
 * whose preimage is $X$.

Proof
As $f: S \to T$ is a mapping, we have that:
 * $\forall x \in S: \left({x, y_1}\right) \in f \land \left({x, y_2}\right) \in f \implies y_1 = y_2$

From the definition of a subset, $x \in X \implies x \in S$, and so:
 * $\forall x \in X: \left({x, y_1}\right) \in f \restriction_X \land \left({x, y_2}\right) \in f \restriction_X \implies y_1 = y_2$

Also from the definition of a mapping:
 * $\forall x \in S: \exists y \in T: \left({x, y}\right) \in f$

Again from the definition of a subset, $x \in X \implies x \in S$, and so:
 * $\forall x \in X: \exists y \in T: \left({x, y}\right) \in f \restriction_X$

So $f \restriction_X: X \to T$ is a mapping.

The fact that the domain of $f \restriction_X$ is $X$ follows from the definition of domain.

The fact that the preimage of $f \restriction_X$ is also $X$ follows from Preimage of Mapping equals Domain.