Mapping is Injection iff Direct Image Mapping is Injection

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

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

Let $f_g$ be an injection.

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

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

Then $\exists x_1 \ne x_2 \in S: g \left({x_1}\right) = g \left({x_2}\right) = y \in Y$.

Let $X_1 = \left\{{x_1}\right\}, X_2 = \left\{{x_2}\right\}, Y = \left\{{y}\right\}$.

Then we see straight away that $f_g \left({X_1}\right) = f_g \left({X_2}\right) = Y$

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

So by the Rule of Transposition, the result follows.