You forget that the biggest driver behind fiat money is the government. As long as taxes are due to the government and they have to be paid in fiat, there will be use for your dollars and euros. No government is going to accept that bitcoins are more valuable than the fiat they print; hence the government will continue to accept fiat money at least for taxes and penalties.
I think most Fiat will lose its value while some government adopt bitcoin with rules / monitoring or prohibit hard activity bitcoin. bitcoin always been a major currencies (among cryptocurrency other), but if you mean major currencies including Fiat, I think it's not going to happen because of the government can only be made regulation or prohibit bitcoin to prevent it happen.