let f1, f2 be Function; :: thesis: ( dom f1 = bool (dom f) & ( for X being set st X c= dom f holds
f1 . X = f .: X ) & dom f2 = bool (dom f) & ( for X being set st X c= dom f holds
f2 . X = f .: X ) implies f1 = f2 )
assume that
A6: dom f1 = bool (dom f) and
A7: for X being set st X c= dom f holds
f1 . X = f .: X and
A8: dom f2 = bool (dom f) and
A9: for X being set st X c= dom f holds
f2 . X = f .: X ; :: thesis: f1 = f2
for x being object st x in bool (dom f) holds
f1 . x = f2 . x

proof

let x be object ; :: thesis: ( x in bool (dom f) implies f1 . x = f2 . x )
assume A10: x in bool (dom f) ; :: thesis: f1 . x = f2 . x
reconsider x = x as set by TARSKI:1;
f1 . x = f .: x by A7, A10;
hence f1 . x = f2 . x by A9, A10; :: thesis: verum

end;

hence f1 = f2 by A6, A8; :: thesis: verum