Technically speaking, the battery isn't powering the converter, just providing an alternate source of 12VDC power. Without 110V AC, the converter is cold and silent.
The converter uses 110V AC power to produce 12V DC power to run lights, water pump etc.
When not plugged into 110V AC power, the 12V DC is provided by a battery, usually a deep cycle type, sometimes called the "house battery." This would NOT be your truck's starting battery which is not designed to be repeatedly discharged.
All the DC appliances are routed to the converter fuse panel. The WCFO 87xx series converters will auto switch between the converter output and the battery output and will also charge your (house) battery when plugged in to 110VAC.
Hope this answers your question.