reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;
reserve e,u for object,
  A for Subset of X;

theorem
  for A,B being set, f being Function st A c= dom f & f.:A c= B holds A c= f"B
proof
  let A,B be set, f being Function;
  assume A c= dom f;
  then
A1: A c= f"(f.:A) by Th75;
  assume f.:A c= B;
  then f"(f.:A) c= f"B by RELAT_1:143;
  hence thesis by A1;
end;
