let f, g be Function; :: thesis: ( g in product f implies rng g c= Union f )
assume A1: g in product f ; :: thesis: rng g c= Union f
then A2: dom g = dom f by CARD_3:18;
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng g or y in Union f )
assume y in rng g ; :: thesis: y in Union f
then consider x being set such that
A3: ( x in dom g & y = g . x ) by FUNCT_1:def 5;
y in f . x by A1, A2, A3, CARD_3:18;
hence y in Union f by A2, A3, CARD_5:10; :: thesis: verum