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;

theorem Th34:
 for x,y being object holds
  {x}=Ch"{y} implies Intersection(F,Ch,y)=F.x
proof let x,y be object;
A1: (dom Ch\{x}) misses {x} by XBOOLE_1:79;
  assume
A2: {x}=Ch"{y};
  then (Ch|(dom Ch\{x}))"{y}=(dom Ch\{x}) /\ {x} by FUNCT_1:70;
  then (Ch|(dom Ch\{x}))"{y}={} by A1;
  then
A3: Intersection(F,Ch|(dom Ch\{x}),y)=union rng F by Th33;
  x in Ch"{y} by A2,TARSKI:def 1;
  then
A4: union rng F/\F.x=Intersection(F,Ch,y) by A3,Th31;
  per cases;
  suppose
    x in dom F;
    then F.x in rng F by FUNCT_1:def 3;
    hence thesis by A4,XBOOLE_1:28,ZFMISC_1:74;
  end;
  suppose
    not x in dom F;
    then F.x={} by FUNCT_1:def 2;
    hence thesis by A4;
  end;
end;
