We present the [CII]158$mu$m line luminosity functions (LFs) at $zsim4-6$ using the ALMA observations of 118 sources, which are selected to have UV luminosity $M_{1500A}<-20.2$ and optical spectroscopic redshifts in COSMOS and ECDF-S. Of the 118 targets, 75 have significant [CII] detections and 43 are upper limits. This is by far the largest sample of [CII] detections which allows us to set constraints to the volume density of [CII] emitters at $zsim4-6$. But because this is a UV-selected sample, we are missing [CII]-bright but UV-faint sources making our constraints strict lower limits. Our derived LFs are statistically consistent with the $zsim0$ [CII] LF at $10^{8.25} - 10^{9.75}L_odot$. We compare our results with the upper limits of the [CII] LF derived from serendipitous sources in the ALPINE maps (Loiacono et al. 2020). We also infer the [CII] LFs based on published far-IR and CO LFs at $zsim4-6$. Combining our robust lower limits with these additional estimates, we set further constraints to the true number density of [CII] emitters at $zsim 4 - 6$. These additional LF estimates are largely above our LF at $L_{[CII]}>10^9L_{odot}$, suggesting that UV-faint but [CII]-bright sources likely make a significant contributions to the [CII] emitter volume density. When we include all the LF estimates, we find that available model predictions underestimate the number densities of [CII] emitters at $zsim4-6$. Finally, we set a constraint on the molecular gas mass density at $zsim4-6$, with $rho_{mol} sim (2-7)times10^7M_odot$,Mpc$^{-3}$. This is broadly consistent with previous studies.
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