reserve X,Y,Z for set,
  x,y,z for object,
  E for non empty set,
  A,B,C for Ordinal ,
  L,L1 for Sequence,
  f,f1,f2,h for Function,
  d,d1,d2,d9 for Element of E;

theorem Th10:
  (dom f = E & for d holds f.d = f.:d) implies rng f is epsilon-transitive
proof
  assume that
A1: dom f = E and
A2: for d holds f.d = f.:d;
  let Y;
  assume Y in rng f;
  then consider b being object such that
A3: b in dom f and
A4: Y = f.b by FUNCT_1:def 3;
  reconsider d = b as Element of E by A1,A3;
A5: f.d = f.:d by A2;
  let a be object;
  assume a in Y;
  then ex c being object st c in dom f & c in d & a = f.c
by A4,A5,FUNCT_1:def 6;
  hence thesis by FUNCT_1:def 3;
end;
