The FAST All Sky H I survey (FASHI) conducted with the Five-hundred-meter Aperture Spherical radio Telescope (FAST) has mapped ~19,500 deg² of the sky north of DEC = -14°, detecting 156,411 extragalactic H I sources at z < 0.09 with a median sensitivity of 0.57 mJy/beam at a velocity resolution of 6.4 km/s. The survey achieves unprecedented depth and area coverage, significantly improving upon previous single-dish surveys. Through a detailed completeness analysis that accounts for the survey's non-uniform sensitivity and line-width dependence, we construct a robust H I mass function (HIMF) using a completeness-corrected sample of over 109,000 sources. The HIMF is robustly constrained down to M_HI ~ 10^6.2 M_sun. When systematic uncertainties are included, the HIMF is well described by a single-Schechter function with a characteristic mass log(M_s / h_70^-2 M_sun) = 9.89 ± 0.02, low-mass end slope α = -1.31 ± 0.02 and amplitude φ_s = (6.38 ± 0.49) × 10^-3 h_70^3 Mpc^-3 dex^-1. The derived cosmic H I density is Ω_HI = (4.71 ± 0.03_stat ± 0.40_sys) × 10^-4 h_70^-1. FASHI provides the most extensive and sensitive H I catalog to date, establishing an important benchmark for studies of gas accretion, galaxy evolution, and large-scale structure in the local universe.