reserve X,Y for set;
reserve G for Group;
reserve n for Nat;

theorem Th12:
  for X,Y being non empty set
  for p being Function of X,Y st p is bijective holds
  SymGroupsIso(p) is onto
  proof
    let X,Y be non empty set;
    let p be Function of X,Y such that
A1: p is bijective;
    set G = SymGroup(X), H = SymGroup(Y);
    set h = SymGroupsIso(p);
A2: dom p = X by FUNCT_2:def 1;
    thus rng h c= the carrier of H;
    let y be object;
    assume y in the carrier of H;
    then reconsider y as Element of H;
    reconsider g = y as Permutation of Y by Th5;
A3: rng p = Y by A1,FUNCT_2:def 3;
    then reconsider p1 = p" as Function of Y,X by A1,FUNCT_2:25;
A4: id Y = p*p1 by A1,A3,FUNCT_2:29;
A5: dom h = the carrier of G by FUNCT_2:def 1;
    set x = p1*g*p;
A6: rng p1 = X by A1,A2,FUNCT_1:33;
    rng g = Y by FUNCT_2:def 3;
    then rng (p1*g) = X by A6,FUNCT_2:14;
    then rng x = X by A3,FUNCT_2:14;
    then x is Permutation of X by A1,FUNCT_2:57;
    then x in permutations X;
    then reconsider x as Element of G by Def2;
    h.x = p*x*p1 by A1,Def3
    .= p*(p1*g)*p*p1 by RELAT_1:36
    .= p*(p1*g)*(p*p1) by RELAT_1:36
    .= (id Y)*g*(id Y) by A4,RELAT_1:36
    .= g*(id Y) by FUNCT_2:17
    .= y by FUNCT_2:17;
    hence thesis by A5,FUNCT_1:def 3;
  end;
