  We make the most precise determination to date of the number density of
extragalactic 21-cm radio sources as a function of their spectral line widths -
the HI velocity width function (HIWF) - based on 21827 sources from the final
7000 deg$^2$ data release of the Arecibo Legacy Fast ALFA (ALFALFA) survey. The
number density of sources as a function of their neutral hydrogen masses - the
HI mass function (HIMF) - has previously been reported to have a significantly
different low-mass slope and 'knee mass' in the two sky regions surveyed during
ALFALFA. In contrast with this, we find that the shape of the HIWF in the same
two sky regions is remarkably similar, consistent with being identical within
the confidence intervals implied by the data (but the overall normalisation
differs). The spatial uniformity of the HIWF implies that it is likely a stable
tracer of the mass function of dark matter haloes, in spite of the
environmental processes to which the measured variation in the HIMF are
attributed, at least for galaxies containing enough neutral hydrogen to be
detected. This insensitivity of the HIWF to galaxy formation and evolution can
be exploited to turn it into a powerful constraint on cosmological models as
future surveys yield increasingly precise measurements. We also report on the
possible influence of a previously overlooked systematic error affecting the
HIWF, which may plausibly see its low-velocity slope steepen by $\sim$40 per
cent in analyses of future, deeper surveys. Finally, we provide an updated
estimate of the ALFALFA completeness limit.
