Let $f:X \rightarrow Y$ be a morphism of projective varieties. If $X=\cup_{i=1}^n X_i$ is the irreducible decomposition of $X$ does the following hold? $$f(X)=f\left(\bigcup_{i=1}^n X_i\right) = \bigcup_{i=1}^n f(X_i)$$
whereby the $f(X_i)$ are the irreducible components of $f(X)$ in the ambient target space $Y$?
My belief is that this is likely to hold due to the preservation of irreducibility of morphisms and the fact that projective morphisms map closed sets onto closed sets. Thank you very much in advance.