Using a rectifier and chopper in place of a transformer
It is possible to provide lower voltage DC from higher voltage AC to DC subject to certain conditions. This capability is what a is generally known as a "Switching Regulator" or "Switched Mode Power Supply" (SMPS).

I'll provide an overview of a subset of the possibiliies as it gets complex" and the following should get you started. Doing a web search on the above terms will tell you much more.

This covers conversion of highr voltage DC to lower voltage DC (or rctification of high voltage AC to DC and then conversion). 

Call the input voltage Vin or HV.
 Call the output voltage Vout or LV. 


A transformer provides isolation against shock or accidental transfer of energy from input to output or output to input in a way which is nit easily achieved by other means. Capacitors can provide isolation, with some important differences.


If you chop a DC power source with an N% duty cycle on/off "switch"  the average voltage delivered to a  resistive load will be N% of the input voltage. IF your load is able to accept peaks of HV N% of the time then you get  alower power level and a lower effective voltage and have effectively eliminated the transformer. Efficiency can be excellent.

However, many loads cannot accept pulses of HV. If you have say 350 V DC s Vin and turn it on 1/70th of the time the effective mean Vout to a resistive load is the same as if you had had Vin/70 = 350/70 = 5V appplied. This may just possibly sound like a USB 5V power supply BUT


Electronic equipment and people do not usually like pulses of 350 VDC instead of continuous 5V DC .
The pulse of HV lead to very large current peaks and then nothing. Thermally the effect is the same as 5VDC continuousas long as you do not ry to actually make 5VDC actual.

If you apply capacitive smoothing to the pulses the current flows through resistors into the capacitor as part of the smoothing action (whether formally inserted ones or wring resistance) and you get massive energy losses. Efficiency is only lv/HV. If LV= 5V and HV = 350V you get 5/350 ~= 1.5% efficiency.
To remedy the above problems you need some sort of lossless energy store to act like an electronic flywheel to take the energy pulses, store the energy in the "flywheel" and then release it across the rest of the cycle when the switch or chopper is off. 
The only electronic element able to do the above is an inductor. There are electromechanical means but they prove to be far less practical than an inductor.
An inductor looks like one winding of a transformer but is essentially far simpler and the basic method of its use is different. In a transformer the majority of the energy transferred is NOT stored in the inductors. In an inductive smmothing system it is.
When a voltage pulse of HV is applied to an inductor the current ioncreases approximately linearly and when the HV is removed the current decreases approximately linearly as energy is given up to the LV circuit.
The resultant circuit is what is known as a "buck converter". It still uses an inductove element but not a transformer. The switch can operate at high frequency allowing the inductor to be far smaller and lighter than a mains based transformer. There is no isolation of input and output.


This circuit shows a basic buck regulator. When the chopper switch is closed energy flows via the inductor to outpit. When the switch is open enrgy continues to flow from th sored emergy in the inductor. The diode provides a path for the energy flow.



A simple practical buck regukator can look like this. The switcyh is inside the IC bewteen Vin and Out.The connection to FB allows the IC to adjust the chopper on/off ratio to get the correct Vout for varying load etc.