Proving $X\sim Y$

Question

Let $f:A\rightarrow B$, a bijection. Suppose $X\subseteq A$ and $Y\subseteq B$ are two sets such that $f(X)\subseteq Y$ and $f^{-1}(Y)\subseteq X$. Show that $X\sim Y$ and $f/X$ is the bijection between them.

Can you help me show it directly without the use of cardinals? On the one hand, it's kinda intuitive claim, but on the other hand I'm having difficulty writting it down.

Thanks.

Answer 1

Hint: the restriction of an injective function to a subset of the domain is still an injective function. Also the inverse of a bijection is still a bijection. Also, Cantor-Berstein.