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 Th49:
  (union rng F)/\X9=union rng Intersect(F,dom F-->X9)
proof
  set I= Intersect(F,dom F-->X9);
  thus (union rng F)/\X9 c= union rng I
  proof
    let x be object such that
A1: x in (union rng F)/\X9;
A2: x in X9 by A1,XBOOLE_0:def 4;
    x in union rng F by A1,XBOOLE_0:def 4;
    then consider Fx be set such that
A3: x in Fx and
A4: Fx in rng F by TARSKI:def 4;
    consider x1 being object such that
A5: x1 in dom F and
A6: F.x1=Fx by A4,FUNCT_1:def 3;
    x1 in dom I by A5,Th48;
    then
A7: I.x1 in rng I by FUNCT_1:def 3;
    I.x1=Fx/\X9 by A5,A6,Th48;
    then x in I.x1 by A3,A2,XBOOLE_0:def 4;
    hence thesis by A7,TARSKI:def 4;
  end;
  thus union rng I c= (union rng F)/\X9
  proof
    let x be object;
    assume x in union rng I;
    then consider Ix be set such that
A8: x in Ix and
A9: Ix in rng I by TARSKI:def 4;
    consider x1 being object such that
A10: x1 in dom I and
A11: I.x1=Ix by A9,FUNCT_1:def 3;
A12: x1 in dom F by A10,Th48;
    then
A13: F.x1 in rng F by FUNCT_1:def 3;
A14: I.x1=F.x1/\X9 by A12,Th48;
    then x in F.x1 by A8,A11,XBOOLE_0:def 4;
    then
A15: x in union rng F by A13,TARSKI:def 4;
    x in X9 by A8,A11,A14,XBOOLE_0:def 4;
    hence thesis by A15,XBOOLE_0:def 4;
  end;
end;
