By performing density functional theory-based first-principles calculations, the formation of a thin monolayer structure, namely InSeF, via fluorination of monolayer InSe is predicted. It is shown that strong interaction of F and In atoms leads to the detachment of In-Se layers in monolayer InSe and 1T-like monolayer InSeF structure is formed. Monolayer InSeF is found to be dynamically stable in terms of its phonon band dispersions. In addition, its Raman spectrum is shown to exhibit totally distinctive features as compared to monolayer InSe. The electronic band dispersions reveal that monolayer InSeF is a direct gap semiconductor whose valence and conduction band edges reside at the $\mathrm{\ensuremath{\Gamma}}$ point. Moreover, the orientation-dependent linear elastic properties of monolayer InSeF are investigated in terms of the in-plane stiffness and Poisson ratio. It is found that monolayer InSeF displays strong in-plane anisotropy in elastic constants and it is slightly softer material as compared to monolayer InSe. Overall, it is proposed that a thin, direct gap semiconducting monolayer InSeF can be formed by full fluorination of monolayer InSe as a new member of the two-dimensional family.