Is it permissible to deal with an Islamic bank in Egypt that operates on the basis of Islamic Mudarabah, but its founders and capital are from usurious banks?

It is obligatory to investigate the matter of the bank. If its transactions are in accordance with Sharia and it is supervised by a trustworthy Sharia supervisory board, then there is no harm in dealing with it. Otherwise, it is impermissible to deal with it, for the consideration is given to the realities of things, not their names.