My professor mentioned a proper mapping theorem after the name of Remmert which says:
Let $X$ and $Y$ be complex manifolds, $f:X \to Y$ be a proper holomorphic map, and $V \subset X$ be a complex analytic subvariety of $ X$, then $f(V)$ is a subvariety of $Y$.
I know this is a deep result, and the proof is not easy, but is there any simple reason which can convince me $f(V)$ is a subvariety, at least intuitively?
Besides, what is the analog result in algebraic geometry? In Hartshorne, he can only say the image is constructible set.
EDIT: One of intuitive explanation which mix the language of complex and algebraic geoemtry may be following: suppose X,Y are projective analytic varieties over $\mathbb{C}$, than they are algebraic varieties. Because proper morphism are closed, and also because the image is a constructible set, it has to be a variety.
The problems of above explanation are: (1) The question is obviously local on Y, but I have to assume X,Y are projective analytic varieties in order to translate back to algebraic varities(GAGA).(2)I still use the result that the image is constructible which is not obvious intuitively,and GAGA to connect analytic variety to algebraic variety. All in all, it is not a good idea to use algebraic geometry to explain complex geometry.