Generalization of Mathematical Induction?

Like Hayden pointed out in the comments, your theorem doesn't really have much to do with mathematical induction. Notice that the first hypothesis of your theorem says that $Y\backslash X \subseteq f(X)$. So the two hypotheses of your theorem combined says that $$Y\backslash X \subseteq f(X) \subseteq X.$$ Your theorem effectively says that if $X\subseteq Y$ and $Y\backslash X\subseteq X$, then $Y=X$. It has nothing to do with $f$ in particular. This is not a statement about induction or even mappings, but rather a statement about set inclusions.

It does not implies induction. One of your hypotheses is that $f(X)\subseteq X$, and when proceeding by induction you want to prove exactly this for $f(n)=n+1$, so you can deduce $X=\mathbb{N}$.