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
A5: dom f1 = f " (SubFuncs (rng f)) and
A6: for x being set st x in f " (SubFuncs (rng f)) holds
f1 . x = proj2 (f . x) and
A7: dom f2 = f " (SubFuncs (rng f)) and
A8: 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 x in f " (SubFuncs (rng f)) ; :: thesis: f1 . x = f2 . x
then ( f1 . x = proj2 (f . x) & f2 . x = proj2 (f . x) ) by A6, A8;
hence f1 . x = f2 . x ; :: thesis: verum

end;

hence f1 = f2 by A5, A7, FUNCT_1:9; :: thesis: verum