So, short answer, he would still be invisible.
Even after failing his save to the grease and falling prone, as invisibility only ends when: the spell's duration ends (for vanish its just a round/caster level), the creature made invisible makes an attack or uses an effect that includes an enemy (unless its as the spell 'greater invisibility'), or the invisibility is dispelled. So he may have still been invisible, but he would have had to spend a move action to get up from prone, then attempted to move out of the grease effect at half speed by making an acrobatics check (thus negating most of the time of invisibility granted him by vanish).
Now, if there was an environmental factor, it could give more clues as to where he was, but he would still be invisible. Take, for example, him falling into a big puddle of mud, there's probably a pretty sizable dent in the puddle where the invisible wizard suddenly lies. you'd still have to roll concealment to hit him, but now you know the area where he is. This is completely dependent on whether your GM wants to say that there are clues as to where he is, but its a pretty safe bet that it would be true