let Z, X1, X2, X3, X4 be set ; :: thesis: ( ( for z being set holds
( z in Z iff ex x1, x2, x3, x4 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & z = [x1,x2,x3,x4] ) ) ) implies Z = [:X1,X2,X3,X4:] )
assume A1: for z being set holds
( z in Z iff ex x1, x2, x3, x4 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & z = [x1,x2,x3,x4] ) ) ; :: thesis: Z = [:X1,X2,X3,X4:]

now

let z be set ; :: thesis: ( ( z in Z implies z in [:[:X1,X2,X3:],X4:] ) & ( z in [:[:X1,X2,X3:],X4:] implies z in Z ) )
thus ( z in Z implies z in [:[:X1,X2,X3:],X4:] ) :: thesis: ( z in [:[:X1,X2,X3:],X4:] implies z in Z )

proof

assume z in Z ; :: thesis: z in [:[:X1,X2,X3:],X4:]
then consider x1, x2, x3, x4 being set such that
A2: ( x1 in X1 & x2 in X2 & x3 in X3 ) and
A3: ( x4 in X4 & z = [x1,x2,x3,x4] ) by A1;
[x1,x2,x3] in [:X1,X2,X3:] by A2, Th73;
hence z in [:[:X1,X2,X3:],X4:] by A3, ZFMISC_1:def 2; :: thesis: verum

end;

assume z in [:[:X1,X2,X3:],X4:] ; :: thesis: z in Z
then consider x123, x4 being set such that
A4: x123 in [:X1,X2,X3:] and
A5: x4 in X4 and
A6: z = [x123,x4] by ZFMISC_1:def 2;
consider x1, x2, x3 being set such that
A7: ( x1 in X1 & x2 in X2 & x3 in X3 ) and
A8: x123 = [x1,x2,x3] by A4, Th72;
z = [x1,x2,x3,x4] by A6, A8;
hence z in Z by A1, A5, A7; :: thesis: verum

end;

then Z = [:[:X1,X2,X3:],X4:] by TARSKI:2;
hence Z = [:X1,X2,X3,X4:] by ZFMISC_1:def 4; :: thesis: verum