Simultaneous localization and mapping (SLAM) plays an important role in autonomous driving, indoor robotics and AR/VR. Outdoor SLAM has been widely used with the assistance of LiDAR and Global Navigation Satellite System (GNSS). However, for indoor applications, the commonly used LiDAR sensor does not satisfy the accuracy requirement and the GNSS signals are blocked. Thus, an accurate and reliable 3D sensor and suited SLAM algorithms are required for indoor SLAM. One of the most promising 3D perceiving techniques, fringe projection profilometry (FPP), shows great potential but does not prevail in indoor SLAM. In this paper, we first introduce FPP to indoor SLAM, and accordingly propose suited SLAM algorithms, thus enabling a new FPP-SLAM. The proposed FPP-SLAM can achieve millimeter-level and real-time mapping and localization without any expensive equipment assistance. The performance is evaluated in both simulated controlled and real room-sized scenes. The experimental results demonstrate that our method outperforms other state-of-the-art methods in terms of efficiency and accuracy. We believe this method paves the way for FPP in indoor SLAM applications.