Apparatus is provided, including: (A) a valve body (204) including a first frame (206) shaped to define a lumen therethrough, and a valve member (205) disposed within the lumen, (B) an upstream support (210), configured to be placed against an upstream surface of a native heart valve, and (C) a flexible sheet (214) that couples the upstream support to the valve body. The valve body has a compressed state in which the first frame has a first diameter, and an expanded state in which the first frame has a second diameter that is greater than the first diameter. The support includes a second frame (212) that has a compressed state, and an expanded state in which the second frame is annular, has an inner perimeter that defines an opening through the second frame, and has an outer perimeter. Other embodiments are also described.