Restriction of Mapping is Mapping/General Result

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

Let $X \subseteq S$.

Let $f \sqbrk X \subseteq Y \subseteq T$.

Let $f \restriction_{X \times Y}$ be the restriction of $f$ to $X \times Y$.

Then $f \restriction_{X \times Y}: X \to Y$ is a mapping:
 * whose domain is $X$
 * whose preimage is $X$
 * whose codomain is $Y$.

$f \restriction_{X \times Y}$ is Many-to-one
We have:

$f \restriction_{X \times Y}$ is Left-total
We have:

Hence by definition, $f \restriction_{X \times Y}: X \to Y$ is a mapping.

By definition of domain, the domain of $f \restriction_{X \times Y}$ is $X$.

By definition of codomain, the codomain of $f \restriction_{X \times Y}$ is $Y$.

From Preimage of Mapping equals Domain, the preimage of $f \restriction_{X \times Y}$ is also $X$.