Image of Class under Mapping is Image of Restriction of Mapping to Class

Theorem
Let $V$ be a basic universe

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

Let $A$ be a class.

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

Then $f \sqbrk A$ is the image of the restriction of $f$ to $A$:
 * $f \sqbrk A = \Img {f {\restriction} A}$

Proof
By definition, $f {\restriction} A$ is class of all ordered pairs $\tuple {a, \map f a}$, where $a \in A$.

Hence $f {\restriction} A$ is a subclass of the cartesian product of $A$ with its image:
 * $f {\restriction} A \subseteq A \times f \sqbrk A$

The result follows directly.