let X be non empty functional compatible set ; :: thesis: rng (union X) = union { (rng f) where f is Element of X : verum }
set F = { (rng f) where f is Element of X : verum } ;

now

let y be set ; :: thesis: ( ( y in rng (union X) implies y in union { (rng f) where f is Element of X : verum } ) & ( y in union { (rng f) where f is Element of X : verum } implies y in rng (union X) ) )

hereby :: thesis: ( y in union { (rng f) where f is Element of X : verum } implies y in rng (union X) )

assume y in rng (union X) ; :: thesis: y in union { (rng f) where f is Element of X : verum }
then consider x being set such that
A1: [x,y] in union X by RELAT_1:def 5;
consider Z being set such that
A2: [x,y] in Z and
A3: Z in X by A1, TARSKI:def 4;
reconsider Z = Z as Element of X by A3;
A4: rng Z in { (rng f) where f is Element of X : verum } ;
y in rng Z by A2, RELAT_1:20;
hence y in union { (rng f) where f is Element of X : verum } by A4, TARSKI:def 4; :: thesis: verum

end;

assume y in union { (rng f) where f is Element of X : verum } ; :: thesis: y in rng (union X)
then consider Z being set such that
A5: y in Z and
A6: Z in { (rng f) where f is Element of X : verum } by TARSKI:def 4;
consider f being Element of X such that
A7: Z = rng f and
verum by A6;
consider x being set such that
A8: [x,y] in f by A5, A7, RELAT_1:def 5;
[x,y] in union X by A8, TARSKI:def 4;
hence y in rng (union X) by RELAT_1:20; :: thesis: verum

end;

hence rng (union X) = union { (rng f) where f is Element of X : verum } by TARSKI:2; :: thesis: verum