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Department of Treasury, so effectively infinite

This is incorrect. It is an insurance scheme paid through bank premiums.

Obviously if the money ran out the Fed would (probably) step in, but that is not how it is structured and supposed to work.

Not "probably". The government is the ultimate backer of the FDIC and would be obligated to step in if FDIC were somehow unable to cover its obligations.

There is no legal obligation for the government to step in. If the FDIC runs out of money, it runs out of money. Everyone expect that IF that were to happen, then Congress would step in and provide legal authorization for the FDIC to use treasury funds, or whatever other scheme is chosen for backing the FDIC's obligations. Because presumably in that situation letting the FDIC fail would be a far worse outcome. But this is not how things are required to play out, with existing legal authorizations. It's just speculation.

Right now, according to the law, if the FDIC rainy day funds run out there is no further source of funds to bail out banks.

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