  We present predictions for number counts and redshift distributions of
galaxies detectable in continuum and in emission lines with the Mid-infrared
(MIR) Instrument (SMI) proposed for the Space Infrared Telescope for Cosmology
and Astrophysics (SPICA). We have considered 24 MIR fine-structure lines, four
Polycyclic Aromatic Hydrocarbon (PAH) bands (at 6.2, 7.7, 8.6 and 11.3$\mu$m)
and two silicate bands (in emission and in absorption) at 9.7$\mu$m and
18.0$\mu$m. Six of these lines are primarily associated with Active Galactic
Nuclei (AGNs), the others with star formation. A survey with the SMI
spectrometers of 1 hour integration per field-of-view (FoV) over an area of
$1\,\hbox{deg}^2$ will yield $5\,\sigma$ detections of $\simeq 140$ AGN lines
and of $\simeq 5.2\times10^{4}$ star-forming galaxies, $\simeq 1.6\times10^{4}$
of which will be detected in at least two lines. The combination of a shallow
($20.0\,\hbox{deg}^{2}$, $1.4\times10^{-1}$ h integration per FoV) and a deep
survey ($6.9\times10^{-3}\,\hbox{deg}^{2}$, $635$ h integration time), with the
SMI camera, for a total of $\sim$1000 h, will accurately determine the MIR
number counts of galaxies and of AGNs over five orders of magnitude in flux
density, reaching values more than one order of magnitude fainter than the
deepest Spitzer $24\,\mu$m surveys. This will allow us to determine the cosmic
star formation rate (SFR) function down to SFRs more than 100 times fainter
than reached by the Herschel Observatory.
