Pullback of Subset Inclusion

Theorem
Denote with $\mathbf{Set}$ the category of sets.

Let $A, B$ be sets, and let $f: A \to B$ be a mapping.

Let $V \subseteq B$ be a subset of $B$, and denote with $i: V \to B$ the inclusion.

Let $f^{-1} \left({V}\right) \subseteq A$ be the preimage of $V$ under $f$; denote with $j: f^{-1} \left({V}\right) \to A$ the inclusion.

Denote with $\bar f = f \restriction_{f^{-1} \left({V}\right)}$ the restriction of $f$ to $f^{-1} \left({V}\right)$.

Then:


 * $\begin{xy}\xymatrix{

f^{-1} \left({V}\right) \ar[r]^*+{\bar f} \ar[d]_*+{j} & V \ar[d]^*+{i}

\\ A \ar[r]_*+{f} & B }\end{xy}$

is a pullback diagram in $\mathbf{Set}$.