Image of Set under Mapping is Set iff Restriction is Set

Theorem
Let $V$ be a basic universe

Let $f: V \to V$ be a mapping.

Let $x$ be a set.

Let $f \sqbrk x$ denote the image of $x$ under $f$.

Let $f {\restriction} x$ denote the restriction of $f$ to $x$.

Then $f \sqbrk x$ is a set $f {\restriction} x$ is a set.