reserve i,j,k,x for object;

theorem Th7:
  for o be set holds FuncComp({id o},{id o}) = (id o,id o) :-> id o
proof
  let o be set;
A1: dom FuncComp({id o},{id o}) = [:{id o},{id o}:] by PARTFUN1:def 2;
  rng FuncComp({id o},{id o}) c= {id o}
  proof
    let i be object;
    assume i in rng FuncComp({id o},{id o});
    then consider j being object such that
A2: j in [:{id o},{id o}:] and
A3: i = FuncComp({id o},{id o}).j by A1,FUNCT_1:def 3;
    consider f,g being Function such that
A4: j = [g,f] and
A5: FuncComp({id o},{id o}).j = g*f by A1,A2,Def9;
    f in {id o} by A2,A4,ZFMISC_1:87;
    then
A6: f = id o by TARSKI:def 1;
    g in {id o} by A2,A4,ZFMISC_1:87;
    then o /\ o = o & g = id o by TARSKI:def 1;
    then i = id o by A3,A5,A6,FUNCT_1:22;
    hence thesis by TARSKI:def 1;
  end;
  then FuncComp({id o},{id o}) is Function of [:{id o},{id o}:],{id o} by A1,
RELSET_1:4;
  hence thesis by FUNCOP_1:def 10;
end;
