Resolution activity may require the temporary use of public money if the Resolution Fund would not have enough resources, for instance, to capitalize a bridge bank that will be sold later on to the private sector thus recovering the capital involved.
In the US this is ensured by the existence of a Treasury credit line that can be drawn upon by the FDIC but that has to be later paid back. This implies then a third element of a complete banking union – the existence of a Financial Backstop that will also be involved in the direct recapitalisation of banks.