Abstract: A prosthetic valve comprises a valve body, comprising a tubular frame and a plurality of leaflets; an upstream support, comprising a second frame; and a flexible sheet that couples the upstream support to the valve body. The prosthetic valve is compressible into a delivery tool such that the prosthetic valve is in a compressed state in which the second frame and the tubular frame are arranged in series within the tool, and the valve body has a compressed body-diameter. The prosthetic valve is deployable from the tool such that the prosthetic valve automatically expands into an expanded state. In the expanded state, the second frame defines an inner perimeter, the sheet extends inwardly from the inner perimeter to the valve body, and the prosthetic valve defines a conduit through the upstream support and the valve body. Other embodiments are also described.