theorem Th88:
  rng R c= Y implies Y|`R = R
proof
  assume rng R c= Y; then
A1: [:dom R,rng R:] c= [:dom R,Y:] by ZFMISC_1:95;
  R c= [:dom R,rng R:] & Y|`R = R /\ [:dom R,Y:] by Th1,Th87;
  hence thesis by A1,XBOOLE_1:1,28;
end;
