
theorem Th14:
  for A be set, B being non empty set, f being Function of A,B
  holds [:f,f:].:id A c= id B
proof
  let A be set, B be non empty set, f be Function of A,B;
  let x be object;
  assume x in [:f,f:].:id A;
  then consider yy being object such that
A1: yy in [:A,A:] and
A2: yy in id A and
A3: [:f,f:].yy = x by FUNCT_2:64;
  consider y,y9 being object such that
A4: y in A and y9 in A and
A5: yy = [y,y9] by A1,ZFMISC_1:84;
A6: y = y9 by A2,A5,RELAT_1:def 10;
  reconsider y as Element of A by A4;
A7: f.y in B by A4,FUNCT_2:5;
A8: y in dom f by A4,FUNCT_2:def 1;
  x = [:f,f:].(y,y9) by A3,A5
    .= [f.y,f.y] by A6,A8,FUNCT_3:def 8;
  hence thesis by A7,RELAT_1:def 10;
end;
