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. We have considered 24 MIR fine-structure lines, four polycyclic aromatic hydrocarbon bands (at 6.2, 7.7, 8.6 and 11.3 μm) and two silicate bands (in emission and in absorption) at 9.7 and 18.0 μm. 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-h integration per field of view (FoV) over an area of 1 deg 2 will yield 5 detections of ~= 140 AGN lines and of ~= 5.2 x 10 4 star-forming galaxies, ~= 1.6 x 10 4 of which will be detected in at least two lines. The combination of a shallow (20.0 deg 2 , 1.4 x 10 –1 h integration per FoV) and a deep survey (6.9 x 10 –3 deg 2 , 635 h integration time), with the SMI camera, for a total of ~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 μm 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.