set Y = { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } ;

A1: { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } c= the carrier of A

then consider I0 being Element of A such that

A2: I0 in ElementaryInstructions A and

A3: for s being Element of Funcs (X,INT) holds f . (s,I0) = s +* ((v . s),(t . s)) ;

I0 in { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } by A2, A3;

then the Element of { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } in { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } ;

hence the Element of { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } is Element of A by A1; :: thesis: verum

A1: { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } c= the carrier of A

proof

v,t form_assignment_wrt f
by Def22;
let i be object ; :: according to TARSKI:def 3 :: thesis: ( not i in { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } or i in the carrier of A )

assume i in { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } ; :: thesis: i in the carrier of A

then ex I being Element of A st

( i = I & I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) ;

hence i in the carrier of A ; :: thesis: verum

end;assume i in { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } ; :: thesis: i in the carrier of A

then ex I being Element of A st

( i = I & I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) ;

hence i in the carrier of A ; :: thesis: verum

then consider I0 being Element of A such that

A2: I0 in ElementaryInstructions A and

A3: for s being Element of Funcs (X,INT) holds f . (s,I0) = s +* ((v . s),(t . s)) ;

I0 in { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } by A2, A3;

then the Element of { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } in { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } ;

hence the Element of { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } is Element of A by A1; :: thesis: verum