CR-MO 2015 G4

Consider $\vartriangle ABC$ , right at $B$ , let $I$ be its incenter and $F,D,E$ the points where the circle inscribed on sides AB, $BC$ and $AC$ , respectively. If $M$ is the intersection point of $CI$ and $EF$ , and $N$ is the intersection point of $DM$ and $AB$ . Prove that $AN = ID$ .