The major fiat currencies use a proof of violence system to establish value. Specifically, engaging in economic activity incurs fiat denominated tax liabilities

The major fiat currencies use a proof of violence system to establish value. Specifically, engaging in economic activity incurs fiat denominated tax liabilities, even if that activity doesn’t use the fiat. The currency issuing and taxing authority has offered proof that it will use whatever violence is necessary to enforce those tax liabilities. This creates a more or less inelastic demand for fiat and thus gives it value.