I want to git rid of unity and all its applications and move to Mate, there are of course a method to install the Mate desktop environment along side Unity, but I don't want that only. I want it to be as If I have Installed the Ubuntu Mate flavour from an ISO file, I mean as if Unity didn't exist at all.
The way you word this question the only way to do this is by re-installing from an Ubuntu Mate ISO are formatting your current installation.
By completly removing Unity you will break your system. By 1st installing Mate and then removing Unity you might break your system. Here is a method for removing Unity where someone wanted Cinnamon installed. That will work for Mate too if change the personal archive to this.
But if those commands are not up to date you will leave traces of Unity on your system. So, to me, a re-install is the only method to meet all your demands.
Ubunutu mate provides nice guide to replace Unity with Mate desktop. http://wiki.mate-desktop.org/replace_unity_by_mate