Are Nikkor DX lenses compatible with Nikon F5?

I'm planning to buy a Nikon F5 and I'd like to know if the lenses I already have (for a Nikon D3000) are compatible with the Nikon F5.

Nikon DX
AF-S NIKKOR 18-55mm
AF-S NIKKOR 55-200mm

The two lenses you named will mount, autofocus, and meter perfectly.

BUT, since they are DX lenses and designed for the smaller DX sensor, you'll get serious vignetting (cutting off of the corners of your image) at the wider lens settings.

Two words, image circle. The DX lenses are designed for APS-C sensors, they give a smaller image circle. But yeah, the lenses will mount and will work, you'll just notice a dark circle (i.e vignetting).

Found this example of the vignetting from a DX lens on a film camera

How serious the vignetting will be will depend on the settings, vignetting will be worse with larger apertures, so if you want to minimise the vignetting, use a larger f-number.