reserve a,b,c for set;

theorem Th31:
  for X being infinite set holds card {F` where F is Subset of X:
  F is finite} = card X
proof
  let X be infinite set;
  set FX = {F` where F is Subset of X: F is finite};
  deffunc f(set) = X\proj2 $1;
  consider f being Function such that
A1: dom f = X* & for a st a in X* holds f.a = f(a) from FUNCT_1:sch 5;
  FX c= rng f
  proof
    let a be object;
    assume a in FX;
    then consider F being Subset of X such that
A2: a = F` and
A3: F is finite;
    consider p being FinSequence such that
A4: F = rng p by A3,FINSEQ_1:52;
    p is FinSequence of X by A4,FINSEQ_1:def 4;
    then
A5: p in X* by FINSEQ_1:def 11;
    then f.p in rng f by A1,FUNCT_1:def 3;
    hence thesis by A1,A2,A4,A5;
  end;
  then card FX c= card (X*) by A1,CARD_1:12;
  hence card FX c= card X by CARD_4:24;
  deffunc f(set) = X\{$1};
  consider f being Function such that
A6: dom f = X & for a st a in X holds f.a = f(a) from FUNCT_1:sch 5;
A7: rng f c= FX
  proof
    let a be object;
    assume a in rng f;
    then consider b being object such that
A8: b in dom f and
A9: a = f.b by FUNCT_1:def 3;
    reconsider b as Element of X by A6,A8;
    {b}` in FX;
    hence thesis by A6,A9;
  end;
  f is one-to-one
  proof
    let a,b be object;
    assume that
A10: a in dom f and
A11: b in dom f and
A12: f.a = f.b;
    reconsider a,b as Element of X by A6,A10,A11;
    {a}` = f.b by A6,A12
      .= {b}` by A6;
    then {a} = {b} by SUBSET_1:42;
    hence thesis by ZFMISC_1:3;
  end;
  hence thesis by A6,A7,CARD_1:10;
end;
