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;
