This question stem from Natural Deduction (FeedBack). The reason why I think it is justifiable to open this up as a separate question is that I am now considering other measures to show it, possibly using the completeness theorem. As stated, my goal is to show:
Injectivity and idempotency implies surjectivity.
As Peter pointed out, the modus pones step is invalid. However, I have great difficulties in finding a way to use the given information in such a way that I can show surjectivity, at least formally. I think I understand it, informally. The imdepotence property restricts (possibly shrinks) the range of the injective function so as it is of the same cardinality as the domain.
The only injective idempotent function $f:X\to X$ is the identity function. Proof: For any $x\in X$, idempotence gives us $f(f(x))=f(x)$. Now apply injectivity (with $f(x)$ in the role of $y$) to infer that $f(x)=x$.