If your house goes into foreclosure, it will usually be sold at auction (i.e., "under the hammer") and the mortgagee (lender) would be entitled (but not required) to bid on it by proffering the amount of the debt owed to him by the mortgagor (debtor) in payment for the house.
If the mortgagee did bid the amount of the debt and won the house, then of course by definition the debt would be fully paid and discharged and no further amount would be owed by the erstwhile mortgagor (debtor) and there would be neither the need nor any justification for the lender to try to reach into the erstwhile debtor's bank account nor to pursue any other of the erstwhile debtor's assets.
If, instead, the mortgagee (lender) did not bid at the auction of the house, and the house should be purchased by another bidder whose bid was in an amount less than the outstanding loan balance secured by the mortgage, then the proceeds of the sale would be paid to the lender and applied as far as they went to the partial payment of the outstanding loan, after which the lender could sue for a so-called deficiency judgment for any balance of the loan still remaining unpaid.
Upon obtaining a deficiency judgment, the lender could pursue any available assets of the debtor to satisfy that judgment, including any bank balances owned by the debtor.
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