I'm not sure it works exactly like that. I think miners have two incentives from supporting the network right now. One is to discover new blocks and get the reward from it, and the other is collecting the fees from the transactions. I think transactions fees will still be collected by miners even if all bitcoins are mined. So the question actually is, will the fees alone be enough for the miners to keep providing their service.
To be honest I think there will be no problem at all. First that will only happen in more than 100 years from now, so two things probably happened by then. One, is that mining hardware have evolved enough, and it's not so costly tu support the network, second, bitcoin price is so high that the fees are more than enough to keep them active.
I lean toward this assessment as being the case in the future as well. If Bitcoin survives that long, the fees will be more than enough to incentivize miners to keep the network secure. Also, the current amount of hash rate securing the network is actually quite excessive, so in the future you only need enough to make it cost prohibitive to keep the attackers at bay. Anyway, as been mentioned, I doubt many of us will still be here in 100 years to worry about it and with 100 years worth of forks in between, it's hard to even imagine what Bitcoin would even look like by then.
