Mapping is Injection if its Direct Image Mapping is Injection

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Theorem

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

Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$.

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.

$\blacksquare$