reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;
reserve S for non empty TopStruct,
  f for Function of T, S,
  H for Subset-Family of S;

theorem Th49:
  for T being 1-sorted, S being non empty 1-sorted, f being Function of T,S
  st rng f = [#]S & f is one-to-one holds dom(f") = [#]S & rng(f") = [#]T
proof
  let T be 1-sorted, S be non empty 1-sorted, f be Function of T,S;
  assume that
A1: rng f = [#]S and
A2: f is one-to-one;
A3: f is onto by A1,FUNCT_2:def 3;
  hence dom(f") = dom((f qua Function)") by A2,Def4
    .= [#]S by A1,A2,FUNCT_1:32;
  thus rng(f") = rng((f qua Function)") by A2,A3,Def4
    .= dom f by A2,FUNCT_1:33
    .= [#]T by FUNCT_2:def 1;
end;
