Restriction of Injection is Injection

Theorem
Let $$f: S \to T$$ be an injection.

Let $$X \subseteq S$$ be a subset of $$S$$.

The restriction of $$f$$ to $$X$$ is also an injection.

Proof
By definition of an injection:


 * $$\forall s_1, s_2 \in S: f \left({s_1}\right) = \left({s_2}\right) \implies s_1 = s_2$$.

Suppose $$f \restriction_X: X \to T$$ were not an injection.

Then $$\exists x_1, x_2 \in X: x_1 \ne x_2, f \left({x_1}\right) = \left({x_2}\right)$$.

But then $$\exists x_1, x_2 \in S: x_1 \ne x_2, f \left({x_1}\right) = \left({x_2}\right)$$.

So $$f: S \to T$$ would not then be an injection.

Hence the result.