Show that a morphism $\varphi:X\rightarrow Y$ is a closed immersion iff $\varphi(|X|)$ is closed in |Y|, $\varphi^*:O_Y\rightarrow\varphi_*O_X$ is a surjection, $|X|\rightarrow\varphi(|X|)$ is a homeomorphism and $\ker\varphi^*$ is of finite type.
We use a quite uncommon definition of closed immersion: $\varphi:X\rightarrow Y$ is a morphism of analytic spaces and there exists a closed analytic subspace Z of Y s.t. $\varphi=\tau\circ\rho$, where $\tau:Z\rightarrow Y$ is the inclusion morphism and $\rho:X\rightarrow Z$ is an isomorphism of analytic spaces.
I feel like "$\rightarrow$" should be clear, but I don't really know where to start.
Note how similar your definition is to the RHS of your iff statement!
Using your definition, what is the image of $\phi$ (using the given $\tau$, $\rho$, and $Z \subset Y$ from your definition)? What you are trying to prove should come almost for free by observing that $\tau$ is an inclusion and $\rho$ an isomorphism of analytic spaces.
By the way, your definition isn't as uncommon as you seem to suggest!