80-400mm sounds like a micro-four-thirds size lens. That makes 160-800mm full frame equivalent, and 120-600mm equivalent on DX crop sensor like the D5500. Am I misreading your question? It's an odd zoom range.
Wow--not understanding the responses. Let me try to rephrase the question. I'm looking for a telephoto zoom lens compatible with a Nikon D5500. It came with a kit lens of 70-300mm. I'm thinking that I might want something a little more powerful for wildlife photography. I see a lot of Nikon AF-S Nikkor Telephoto Zoom 80-400mm F/4.5-5.6 G ED VR in the marketplace--used ones at reasonable prices, but I don't think this would be fully compatible with the Nikon D5500 camera body correct?
F mount with an internal autofocus motor, so it will work on a D5500.
OK, I see now. It's a lens I never really paid attention too. Reason being is that for a lot less money you can pick up the Tamron or Sigma 150-600mm lens. Bigger reach, good optical behavior, way less expensive. The downsides being the lenses are larger and heavier, and out at 600mm the minimum aperture is f/6.3, but at 400mm it is f/5.6 like the Nikon lens. They are regarded as good wildlife lenses for a hobbyist.
FX is a camera body with a full-frame sensor, meaning that the sensor is the same size as 35mm film.
DX is a camera body with an APS-C sensor. This is the equivalent size of the Kodak Advantix "Advanced Photo System" film from the 1990's. It is smaller in size than 35mm and it "crops" the image projected on it from the lens. Commonly called a crop-sensor. The field of view is not as wide, and you have to multiply the zoom range by 1.5x to get the FX equivalent focal length. Your D5500 is this. It also does not have the camera-body autofocus motor, so it will not autofocus with older lenses that do not have an internal motor. But otherwise it works with most lenses.
So a Sigma 150-600mm Contemporary lens will work fine with your camera (as will the 80-400mm AF-S by Nikon). It will also give you the 1.5x crop factor, so you would have a lens equivalent to 225mm-900mm in range. Pretty nice for wildlife.
The short answer to your question is yes, an FX lens will work fine on your D5500.
Another option to consider is the Nikon 200-500 F5.6. It has an excellent reputation for image quality and is a constant F5.6 aperture over the entire zoom range. It runs $1400 new, which is considerably less than an 80-400, and frequently is on sale for about $1050. On sale it runs a little more than the Sigma 150-600, but has some advantages over the Sigma, which has a good reputation nonetheless. I have used my 200-500 on a crop frame D3400 and it delivered excellent results.