
theorem Th3:
  for X being trivial set, x being set st x in X for f being
  Function of X,X holds x is_a_fixpoint_of f
proof
  let X be trivial set, x be set;
  assume
A1: x in X;
  then consider y being object such that
A2: X = {y} by ZFMISC_1:131;
  let f be Function of X,X;
  thus x in dom f by A1,FUNCT_2:52;
  then
A3: f.x in rng f by FUNCT_1:def 3;
A4: rng f c= X by RELAT_1:def 19;
  thus x = y by A1,A2,TARSKI:def 1
    .= f.x by A2,A3,A4,TARSKI:def 1;
