Yes the fiat currency would continue to exist except maybe in future, different government decided to embrace Bitcoin as their legal currency but I doubt if it would be possible cause of how volatile it is and won't be suitable for carrying out transactions by market sellers and business owners anyways Bitcoin was to serve as an alternative to fiat against third party but currently it's more like a store of value and people would rather hold it for years to be more valuable than use it for carrying out daily transactions like the fiat currency.
Dude even in the future, fiat system will never go into extinction, that's traditional money we are taking about although I understand why you said in the future, may be you said that because we don't know what the future will be like but I believe that Bitcoin and fiat will continue to exist separately because both currency serves different purposea, people will continue to invest in Bitcoin, they'll take it for what it is but as for fiat itself, it has its own importance, let's also know that Bitcoin is not handy yet and we don't know if such will be possible in the future, Bitcoin is a store of wealth and value for the future, there many reasons why hodlers wouldn't want to transact with their Bitcoin on daily bases just as you said.