reserve x, x1, x2, y, z, X9 for set,
  X, Y for finite set,
  n, k, m for Nat,
  f for Function;
reserve F,Ch for Function;
reserve Fy for finite-yielding Function;

theorem Th50:
  Intersection(F,Ch,y)/\X9= Intersection(Intersect(F,dom F-->X9), Ch,y)
proof
  set I=Intersect(F,dom F-->X9);
  set Int1=Intersection(F,Ch,y);
  set Int2=Intersection(I,Ch,y);
  thus Int1/\X9 c= Int2
  proof
    let x be object such that
A1: x in Int1/\X9;
A2: for z st z in dom Ch & Ch.z=y holds x in I.z
    proof
A3:   x in Int1 by A1,XBOOLE_0:def 4;
      let z;
      assume z in dom Ch & Ch.z=y;
      then
A4:   x in F.z by A3,Def2;
      then
A5:   z in dom F by FUNCT_1:def 2;
      x in X9 by A1,XBOOLE_0:def 4;
      then x in F.z/\X9 by A4,XBOOLE_0:def 4;
      hence thesis by A5,Th48;
    end;
    x in X9 by A1,XBOOLE_0:def 4;
    then x in (union rng F)/\X9 by A1,XBOOLE_0:def 4;
    then x in union rng I by Th49;
    hence thesis by A2,Def2;
  end;
  thus Int2 c= Int1/\X9
  proof
    let x be object such that
A6: x in Int2;
    x in union rng I by A6;
    then
A7: x in (union rng F)/\X9 by Th49;
    then
A8: x in X9 by XBOOLE_0:def 4;
A9: for z st z in dom Ch & Ch.z=y holds x in F.z
    proof
A10:  dom I=dom F by Th48;
      let z;
      assume z in dom Ch & Ch.z=y;
      then
A11:  x in I.z by A6,Def2;
      then z in dom I by FUNCT_1:def 2;
      then x in F.z/\X9 by A11,A10,Th48;
      hence thesis by XBOOLE_0:def 4;
    end;
    x in union rng F by A7,XBOOLE_0:def 4;
    then x in Int1 by A9,Def2;
    hence thesis by A8,XBOOLE_0:def 4;
  end;
end;
