If Mairsil gains an activated ability that's normally linked to another activated ability of the card it came from, those two abilities Mairsil gains are linked for as long as Mairsil remains on the battlefield. For example, if Mairsil exiles Izzet Chemister, cards exiled with the first ability Mairsil gained from Izzet Chemister can be cast if you activate the second ability it gained that way. Izzet Chemister itself can't be cast this way, and if Mairsil leaves the battlefield before activating that second ability, the cards exiled with that first ability are lost forever.