let X, Y be set ; :: thesis: for f being Function st X c= dom f & f .: X c= Y holds
<:f,X,Y:> is total
let f be Function; :: thesis: ( X c= dom f & f .: X c= Y implies <:f,X,Y:> is total )
assume that
A1: X c= dom f and
A2: f .: X c= Y ; :: thesis: <:f,X,Y:> is total
X c= dom <:f,X,Y:>

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in dom <:f,X,Y:> )
assume A3: x in X ; :: thesis: x in dom <:f,X,Y:>
then f . x in f .: X by A1, FUNCT_1:def 12;
hence x in dom <:f,X,Y:> by A1, A2, A3, Th78; :: thesis: verum

end;

hence dom <:f,X,Y:> = X by XBOOLE_0:def 10; :: according to PARTFUN1:def 4 :: thesis: verum