If the fdic insures money in each account for a bank, and that bank goes under. Won't that mean the fdic will have to borrow more money from the fed and thus lead to substantial inflation?
No, for three reasons. First, FDIC bailouts are funded via the guarantee fund, which is a pre-funded insurance fund that has never required an injection of cash. Second, if such an injection were to occur, it would happen through an appropriation from Congress, which would be implemented via a debt issuance from the Treasury— the Fed has zero authority to bail out the FDIC guarantee fund. Third, bank failures actually tend to involve a net contraction of the money supply, as distressed banks are less able to engage in new credit extension.