let f1, f2 be Function; :: thesis: ( dom f1 = f " (SubFuncs (rng f)) & ( for x being set st x in f " (SubFuncs (rng f)) holds
f1 . x = proj2 (f . x) ) & dom f2 = f " (SubFuncs (rng f)) & ( for x being set st x in f " (SubFuncs (rng f)) holds
f2 . x = proj2 (f . x) ) implies f1 = f2 )
assume that
A6: dom f1 = f " (SubFuncs (rng f)) and
A7: for x being set st x in f " (SubFuncs (rng f)) holds
f1 . x = proj2 (f . x) and
A8: dom f2 = f " (SubFuncs (rng f)) and
A9: for x being set st x in f " (SubFuncs (rng f)) holds
f2 . x = proj2 (f . x) ; :: thesis: f1 = f2

now

let x be set ; :: thesis: ( x in f " (SubFuncs (rng f)) implies f1 . x = f2 . x )
assume A10: x in f " (SubFuncs (rng f)) ; :: thesis: f1 . x = f2 . x
then f1 . x = proj2 (f . x) by A7;
hence f1 . x = f2 . x by A9, A10; :: thesis: verum

end;

hence f1 = f2 by A6, A8, FUNCT_1:2; :: thesis: verum