Mapping is Injection iff Direct Image Mapping is Injection

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

Let $f^\to: \powerset S \to \powerset T$ be the mapping induced on $\powerset S$ 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:
 * $\exists x_1 \ne x_2 \in S: \map f {x_1} = \map f {x_2} = y$

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

Then it follows that:
 * $\map {f^\to} {X_1} = \map {f^\to} {X_2} = Y$

Thus $f^\to: \powerset S \to \powerset T$ is not injective.

So by the Rule of Transposition, the result follows.