There is some truth to that, but deposits from savers make up a really insignificant part of the capital from which banks are able to extend credit. Deposits are really now effectively divorced from lending and banks provide for deposits mostly just as a service to customers who want to preserve liquid savings, hence the higher fees for providing that service today.  

The capital that supports lending comes from large capital market operations, such as selling bank stock, bonds, or other securities to finance lending.  In modern capital markets, banks don't need depositors at all, because they can get all the capital they need from very large investors.  The only reason banks have depositors is because governments force them to do so to gain some of the legal and regulatory advantages of being a bank.  For example, Goldman Sachs became a deposit-taking bank in order to benefit from Fed bailout funds during this past crisis.