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 Th53:
  x in dom F implies union rng F=union rng (F|(dom F\{x}))\/F.x
proof
  set d=dom F\{x};
  set Fd=F|d;
A1: F|dom F=F;
  assume
A2: x in dom F;
  then d\/{x}=dom F by ZFMISC_1:116;
  then F=Fd\/(F|{x}) by A1,RELAT_1:78;
  then
A3: rng F= rng Fd\/rng (F|{x}) by RELAT_1:12;
  Im(F,x)={F.x} by A2,FUNCT_1:59;
  then rng (F|{x})={F.x} by RELAT_1:115;
  then union rng F=union rng Fd \/union {F.x} by A3,ZFMISC_1:78;
  hence thesis by ZFMISC_1:25;
end;
