
theorem Th11:
  for A,B be set, f being Function of A,B st f is onto
  holds id B c= [:f,f:].:id A
proof
  let A,B be set, f be Function of A,B;
  assume f is onto;
  then
A1: rng f = B;
  let xx be object;
  assume
A2: xx in id B;
  then consider x,x9 being object such that
A3: xx = [x,x9] by RELAT_1:def 1;
A4: x = x9 by A2,A3,RELAT_1:def 10;
A5: x in B by A2,A3,RELAT_1:def 10;
  then consider y being object such that
A6: y in A and
A7: f.y = x by A1,FUNCT_2:11;
A8: dom f = A by A5,FUNCT_2:def 1;
A9: [y,y] in id A by A6,RELAT_1:def 10;
  [:f,f:].(y,y) = xx by A3,A4,A6,A7,A8,FUNCT_3:def 8;
  hence thesis by A2,A9,FUNCT_2:35;
end;
