Using upper percentile points....

Which means that

we have with a=.95............

(Note that is a rv and is a percentile point.)

Taking the reciprocal inside the probability we have

(Which follows by the definition of an F, it's a ratio of chi-squares divided by their dfs.)

And note that

also.

Thus by the continuity of the densities, these percentile points must be equal....

Hence must equal

and this is true for any probability, not just .05.