Introducing the concept of function into inverse P-sets (inverse packet sets) and improving it, function inverse P-sets (function inverse packet sets) is obtained. Function inverse P-sets is the function set pair composed of function internal inverse P-set (function internal inverse packet set)$\bar{S}^{F}$and function outer inverse P-set (function outer inverse packet set) $\bar{S}^{F}$, or ($\bar{S}^{F}$,$\bar{S}^{F}$ )is function inverse P-sets. Function inverse P-sets, which have dynamic characteristic and law characteristic (or function characteristic), can be reduced to finite general function sets S under certain condition. Inverse P-sets is obtained by introducing dynamic characteristic to finite general element setX (Cantor setX) and improving it. Inverse P-sets is the element set pair composed of internal inverse P-set $\bar{X}^{F}$(internal inverse packet set$\bar{X}^{F}$) and outer inverse P-set $\bar{X}^{\bar{F}}$(outer inverse packet set $\bar{X}^{\bar{F}}$), or $\bar{X}^{F}$,$\bar{X}^{F}$is inverse P-sets which has dynamic characteristic. In this paper, the structure of function inverse P-sets and its reduction, the inverse P-information law fusion generated by function inverse P-sets, and the attribute characteristics and attribute theorems of inverse P-information law are proposed. Using these theoretical results, the hiding image and its applications generated by inverse P-information law fusion are given, which is one of the important applications of function inverse P-sets.