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.