Yesterday one of my friend and I was discussing about the Principle of Mathematical Induction. After some time he made the following,
Theorem. Let $\emptyset \subset X\subseteq Y$ and $f:X\to Y$ be a function such that,
for all $y\in Y\setminus X$ there exists $x\in X$ such that $f(x)=y$.
$f(X)\subseteq X$.
Then $X=Y$.
Sketch of the Proof of the Claim. If $X=Y$ then we have nothing to prove. Otherwise suppose that $X\subset Y$. Then there exists $y\in Y\setminus X$. But by the first property we can conclude that there exists $x\in X$ such that $f(x)=y$. By the second property we can conclude that $y(=f(x))\in X$, a contradiction. So we are done.
Now the interesting thing is that from the above theorem we can prove the so called Weak Principle of Mathematical Induction by taking $Y=\mathbb{N}$ and $X\subseteq \mathbb{N}$ such that $1\in X$ and by taking $f:X\to \mathbb{N}$ defined by $f(n)=n+1$ for all $n\in X$.
The above observation suggests that the above theorem is more general than the Principle of Mathematical Induction. But I couldn't find any literature regarding this type of 'induction'. The question is,
Is this type of 'induction' well known? If so, can some literature regarding this be mentioned?
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.