Uber also is currently much more expensive than car ownership, and if a relatively low-cost driverless car gets on the market, there is literally no incentive for most people to ever use Uber rather than owning their own driverless car.
So in the event of driverless cars, Uber will have to slash prices -- and, unfortunately for them, a self driving car does not in fact claw back all of their driver's revenue.
Now, there are plausible scenarios (though it's by no means guaranteed) in which driverless cars vastly increase the size of the rides-for-hire market, so even if Uber has to take their current margin on lower-cost rides and pay for inventory, you could imagine that working out overall well for them.
But if it's such a great business to be in, why won't they face competition? There are plenty of companies that are deeper-pocketed than Uber. There are in fact plenty of companies with stronger brands than Uber. And the geographical opportunity to compete locally is pretty brutal.
What's Uber going to compete on?
Size of inventory? That drives up costs (when we're already stipulating that they are paying billions in new costs).
Price? That drives down revenue (when we're already stipulating that they're slashing prices).
Cleanliness, as various people have suggested? That means they have to get into a whole new hyper-local physical business with its own logistical problems, and why should we imagine that Uber can build the best car-cleaning business of its field? What about Uber makes us believe they're the best at that? Is this even a field that can be usefully competed on? Are people willing to pay top dollar for a really really really clean car, or does everyone bring together a basic cleaning service and, whatever, it's all good to the passengers?