consider X being set such that

A1: for x being object holds

( x in X iff ( x in Funcs ( the carrier of V, the carrier of V) & S_{1}[x] ) )
from XBOOLE_0:sch 1();

take X ; :: thesis: for x being object holds

( x in X iff x is onto isometric Function of V,V )

let x be object ; :: thesis: ( x in X iff x is onto isometric Function of V,V )

thus ( x in X implies x is onto isometric Function of V,V ) by A1; :: thesis: ( x is onto isometric Function of V,V implies x in X )

assume A2: x is onto isometric Function of V,V ; :: thesis: x in X

then x in Funcs ( the carrier of V, the carrier of V) by FUNCT_2:8;

hence x in X by A1, A2; :: thesis: verum

A1: for x being object holds

( x in X iff ( x in Funcs ( the carrier of V, the carrier of V) & S

take X ; :: thesis: for x being object holds

( x in X iff x is onto isometric Function of V,V )

let x be object ; :: thesis: ( x in X iff x is onto isometric Function of V,V )

thus ( x in X implies x is onto isometric Function of V,V ) by A1; :: thesis: ( x is onto isometric Function of V,V implies x in X )

assume A2: x is onto isometric Function of V,V ; :: thesis: x in X

then x in Funcs ( the carrier of V, the carrier of V) by FUNCT_2:8;

hence x in X by A1, A2; :: thesis: verum