reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;

theorem Th7:
  X <> {} implies Union (X --> Y) = Y
proof
  assume
A1: X <> {};
  set x = the Element of X;
  thus Union (X --> Y) c= Y by Th6;
A2: dom (X --> Y) = X;
  (X --> Y).x = Y by A1,FUNCOP_1:7;
  then Y in rng (X --> Y) by A1,A2,FUNCT_1:def 3;
  hence thesis by ZFMISC_1:74;
end;
