let y, X1, X2, X3, X4, X5 be set ; :: thesis: ( y in [:X1,X2,X3,X4,X5:] implies ex x1, x2, x3, x4, x5 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & y = [x1,x2,x3,x4,x5] ) )
assume y in [:X1,X2,X3,X4,X5:] ; :: thesis: ex x1, x2, x3, x4, x5 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & y = [x1,x2,x3,x4,x5] )
then consider x1234, x5 being set such that
A1: x1234 in [:X1,X2,X3,X4:] and
A2: x5 in X5 and
A3: y = [x1234,x5] by ZFMISC_1:def 2;
consider x1, x2, x3, x4 being set such that
A4: ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 ) and
A5: x1234 = [x1,x2,x3,x4] by A1, MCART_1:79;
y = [x1,x2,x3,x4,x5] by A3, A5;
hence ex x1, x2, x3, x4, x5 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & y = [x1,x2,x3,x4,x5] ) by A2, A4; :: thesis: verum