You should be a little careful with the 10 V, though. Smoothing it after rectification will give you somewhere between 11.5 and 12 V when loaded and a little more than 12 V unloaded.
It is because of the peak rectification that takes place and since the peak voltage is sqrt(2) higher than the RMS value you will have a higher than expected voltage. So be prepared to adjust the turns ratio of the transformer.
Or use 12 V AC and put a linear regulator after the capacitors. Normal regulators need at least 3 V to "drop off" so you can not use 10 V AC. You need a high current regulator and a good heat sink if the load is 3 A.