yes it will work, as long you use a 7700 inner chain ring. Otherwise you might encounter some skating due to the smaller 9-speed chain.
When the move from 8- to 9-speed took place, starting with 1997 Dura-Ace, a very slight change in chainring design occurred. Shimano was concerned about the potential for the chain to "skate" over the teeth of the small chainring.
"Skating" occurs when a narrow chain is used on a crankset intended for a somewhat wider chain. In downshifting, instead of the chain meshing properly with the teeth of the inner chainring, the side-plates of the chain can ride along the tips of the teeth of the small ring, causing a momentary "freewheeling" forward. If you are foolish enough to downshift the front while standing up, this skating could conceivably cause you to crash.
The difference between "9-speed" cranksets and older models is only in the inner chainring. The teeth on a "9-speed" inner ring are slightly displaced to the right to better accommodate the slightly narrower chain.
Shimano will tell you you need to replace the inner ring when converting an older bike to 9-speed. Their lawyers say that covers them if you're clumsy enough to hurt yourself due to "skating" and, besides, they make a nice profit selling the chainrings.
My advice is to not worry about this, and not to routinely replace the inner ring when upgrading to 9-speed. This type of "skating" is very rare, and mostly only happens if you're in top gear front and rear, then decide to downshift the front before downshifting the rear. There's no situation where this is a rational shifting sequence anyway.
Skating can also happen when you don't properly mount the inner chain ring. Since the tooth are not exactly in the middle, mounting the rings in the wrong way will cause additional spacing between the outer and the inner chain ring. This spacing is enough for the chain to slip between them.