# Appropriate family with connected fibers on a connected schema

I was reading Q. Liu's handbook and there is an exercise:

Let $$f: X rightarrow Y$$ to be a surjective morphism. We suppose that $$Y$$ is connected
and that all the fibers $$X_y$$ is connected. Show that if $$X$$ is correct on $$Y$$ then $$X$$ is connected.

I have some ideas but not a complete proof. If we have an arbitrary ringed space $$(X, O_X)$$ and there are no non-trivial idempotents in $$O_X (X)$$, then the space is connected. Then the Stacks 09V4 project tag implies that if we have a proper morphism $$f: X rightarrow Y$$then $$(f_ * O_X) _y = O_X (f ^ {- 1} (y))$$ for any point $$y in Y$$. This combination with fiber connectivity implies that $$(f_ * O_X) _y$$ does not contain any non-trivial idempotent for any $$y in Y$$. I probably should somehow use the direct definition of the stem but I do not see how.