Injection between power sets

Statement

Let $A$ and $B$ be sets with $A \subseteq B$. Prove that there is an injective function from $\mathcal{P}(A)$ into $\mathcal{P}(B)$.

Solution

Define $\iota : \mathcal{P}(A) \to \mathcal{P}(B)$ by $\iota(X) = X$ (each subset of $A$ viewed as a subset of $B$).

It is well defined: if $X \in \mathcal{P}(A)$ then $X \subseteq A$; since $A \subseteq B$, by transitivity $X \subseteq B$, i.e. $X \in \mathcal{P}(B)$.

It is injective: if $\iota(X_1) = \iota(X_2)$ then $X_1 = X_2$ as sets, trivially. So $\iota$ is injective.

Therefore $\mathcal{P}(A)$ embeds into $\mathcal{P}(B)$. $\blacksquare$