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 $\mathbf{Set}$.

Proof
From the definition of pullback, given a commutative diagram:


 * $\begin{xy}\xymatrix{

Q \ar[r]^*+{q_1} \ar[d]_*+{q_2} & V \ar[d]^*+{i}

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

we need to find a mapping $u: Q \to f^{-1} \left({V}\right)$ such that $\bar f \circ u = q_1$ and $j \circ u = q_2$.

Let $x \in Q$; then $q_1 \left({x}\right) \in V \subseteq B$.

Note that the commutativity of the diagram implies that $q_1 \left({x}\right) = f \left({q_2 \left({x}\right)}\right)$ since $i$ is an inclusion.

Hence also $f \left({q_2 \left({x}\right)}\right) \in V$.

It follows by definition of preimage that $q_2 \left({x}\right) \in f^{-1} \left({V}\right)$.

Thus define $u: Q \to f^{-1} \left({V}\right)$ by restriction of $q_2$:


 * $u \left({x}\right) := q_2 \restriction_{Q \times f^{-1} \left({V}\right)} \left({x}\right)$

Then we have:


 * $j \circ u \left({x}\right) = u \left({x}\right) = q_2 \left({x}\right)$

since $j$ is an inclusion mapping.

Hence $q_2 = j \circ u$ by Equality of Mappings.

Furthermore:


 * $\bar f \circ u \left({x}\right) = \bar f \circ q_2 \left({x}\right) = f \left({q_2 \left({x}\right)}\right) = q_1 \left({x}\right)$

since $\bar f$ and $u$ are restrictions of $f$ and $q_2$, respectively.

Hence $\bar f \circ u = q_1$ by Equality of Mappings.

Observe that any $u': Q \to f^{-1} \left({V}\right)$ must satisfy:


 * $j \circ u' = q_2 = j \circ u$

By Inclusion Mapping is Injection, $j$ is an injection.

From Injection iff Monomorphism in Category of Sets, we conclude $j$ is a monomorphism, and so $u = u'$.

This establishes that $u: Q \to f^{-1} \left({V}\right)$ is unique.

Therefore, we have established that:


 * $\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 $\mathbf{Set}$.