Is always true that if $f$ is a bijection from two sets $A$ and $B$, and $C \subset A$, then the restriction $f{\restriction_{C}}$ must be injective?
I've tried proving it by noticing that saying this, is the same as saying that if $f$ is our bijection, and $\iota_{C} : C \to A : x \mapsto x$ is the inclusion application of $C$ in $A$, then $f \circ \iota_{C}$ is injective, but i didn't gone too far, and I'm not even sure of the correctness of the last result.
Yes, the restriction of any injection itself is an injection. For otherwise there would be $x \neq y$, $x,y \in C$ such that $$ f \restriction C (x) = f \restriction C (y). $$ But then $f(x) = f(y)$ -- contradicting the fact that $f$ is injective.