Mapping is Injection iff Direct Image Mapping is Injection

Theorem
Let $f: S \to T$ be a mapping.

Let $f^\to: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ be the mapping induced on $\mathcal P \left({S}\right)$ by $f$.

Let $f^\to$ be an injection.

Then $f: S \to T$ is also an injection.

Proof
Suppose $f: S \to T$ is a mapping, but not injective.

Then:
 * $\exists x_1 \ne x_2 \in S: f \left({x_1}\right) = f \left({x_2}\right) = y$

Let:
 * $X_1 = \left\{{x_1}\right\}$
 * $X_2 = \left\{{x_2}\right\}$
 * $Y = \left\{{y}\right\}$

Then it follows that:
 * $f^\to \left({X_1}\right) = f^\to \left({X_2}\right) = Y$

Thus $f^\to: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ is not injective.

So by the Rule of Transposition, the result follows.