Locally connected and compact Hausdorff space invariant of continuous mappings

I am looking for a reference (not proof) to the following theorem:

If $X$ is a compact and locally connected topological space, Y is a Hausdorff topological space, $f:X\to Y$ is continuous and $f(X)=Y$, then $Y$ is locally connected.

I found reference in Kuratowski, Topology II for the case where $X$ is metrizable. Any one knows about any reference to this case?