rng f c/= {{}} by RELAT_1:def 15;
then consider x being object such that
A1: ( x in rng f & x nin {{}} ) ;
reconsider x = x as set by TARSKI:1;
( x <> {} & x c= Union f ) by A1, TARSKI:def 1, ZFMISC_1:74;
hence not Union f is empty ; :: thesis: verum