reserve x,y,X,Y for set;
reserve C,D,E for non empty set;
reserve SC for Subset of C;
reserve SD for Subset of D;
reserve SE for Subset of E;
reserve c,c1,c2 for Element of C;
reserve d,d1,d2 for Element of D;
reserve e for Element of E;
reserve f,f1,g for PartFunc of C,D;
reserve t for PartFunc of D,C;
reserve s for PartFunc of D,E;
reserve h for PartFunc of C,E;
reserve F for PartFunc of D,D;

theorem Th26:
  SC = f"X iff for c holds c in SC iff c in dom f & f/.c in X
proof
  thus SC = f"X implies for c holds c in SC iff c in dom f & f/.c in X
  proof
    assume
A1: SC = f"X;
    let c;
    thus c in SC implies c in dom f & f/.c in X
    proof
      assume c in SC;
      then c in dom f & (f qua Function).c in X by A1,FUNCT_1:def 7;
      hence thesis by PARTFUN1:def 6;
    end;
    assume that
A2: c in dom f and
A3: f/.c in X;
    (f qua Function).c in X by A2,A3,PARTFUN1:def 6;
    hence thesis by A1,A2,FUNCT_1:def 7;
  end;
  assume
A4: for c holds c in SC iff c in dom f & f/.c in X;
  now
    let x be object;
    thus x in SC implies x in dom f & (f qua Function).x in X
    proof
      assume
A5:   x in SC;
      then reconsider x1=x as Element of C;
      x1 in dom f & f/.x1 in X by A4,A5;
      hence thesis by PARTFUN1:def 6;
    end;
    assume that
A6: x in dom f and
A7: (f qua Function).x in X;
    reconsider x1=x as Element of C by A6;
    f/.x1 in X by A6,A7,PARTFUN1:def 6;
    hence x in SC by A4,A6;
  end;
  hence thesis by FUNCT_1:def 7;
end;
