Let define . I do not like to use because it does not usually mean the power set, instead I shall use for the power set. Now define by . Assume that is one-to-one and then . To show is one-to-one we need to show . Thus, let then . Since it means and so by definition for some . However, is one-to-one so . Thus, , repeating the mirror argument shows , which means . Let us summarize what we did: we showed that if is one-to-one then is one-to-one.

Conversely, say is one-to-one, i.e. if then . We want to show is one-to-one. Say that . Now form the sets and which are subsets of . Note and . Therefore but by above assumptions it means and so . Let us summarize what we did: we showed that if is one-to-one then is one-to-one.

Putting these together it means is one-to-one if and only if is one-to-one.

You try the second part.