 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem ThSubFamIsBoolValued:
  for I being set
  for G being Group
  for F being Subgroup-Family of I,G
  holds Carrier F is (bool the carrier of G)-valued
proof
  let I be set;
  let G be Group;
  let F be Subgroup-Family of I,G;
  per cases;
  suppose I is empty;
    then A1: Carrier F = {};
    rng {} = {} & {} c= (bool the carrier of G) by XBOOLE_1:2;
    hence Carrier F is (bool the carrier of G)-valued by A1,RELAT_1:def 19;
  end;
  suppose I is non empty;
    then reconsider I as non empty set;
    reconsider F as Subgroup-Family of I,G;
    for z being object st z in rng (Carrier F)
    holds z in bool the carrier of G
    proof
      let z be object;
      assume A3: z in rng (Carrier F);
      reconsider y=z as set by TARSKI:1;
      consider i being object such that
      A4: i in dom (Carrier F) & y = (Carrier F).i
      by A3, FUNCT_1:def 3;
      reconsider i as Element of I by A4;
      y = the carrier of F.i by A4, Th9;
      then y c= the carrier of G by GROUP_2:def 5;
      hence thesis by ZFMISC_1:def 1;
    end;
    hence thesis by RELAT_1:def 19, TARSKI:def 3;
  end;
end;
