Theorem setsv

Description: The value of the structure replacement function is a set. (Contributed by AV, 10-Nov-2021)

	Ref	Expression
Assertion	setsv	|- ( ( S e. V /\ B e. W ) -> ( S sSet <. A , B >. ) e. _V )

Proof

Step	Hyp	Ref	Expression
1		setsval	|- ( ( S e. V /\ B e. W ) -> ( S sSet <. A , B >. ) = ( ( S |` ( _V \ { A } ) ) u. { <. A , B >. } ) )
2		resexg	|- ( S e. V -> ( S |` ( _V \ { A } ) ) e. _V )
3		snex	|- { <. A , B >. } e. _V
4	3	a1i	|- ( ( S e. V /\ B e. W ) -> { <. A , B >. } e. _V )
5		unexg	|- ( ( ( S |` ( _V \ { A } ) ) e. _V /\ { <. A , B >. } e. _V ) -> ( ( S |` ( _V \ { A } ) ) u. { <. A , B >. } ) e. _V )
6	2 4 5	syl2an2r	|- ( ( S e. V /\ B e. W ) -> ( ( S |` ( _V \ { A } ) ) u. { <. A , B >. } ) e. _V )
7	1 6	eqeltrd	|- ( ( S e. V /\ B e. W ) -> ( S sSet <. A , B >. ) e. _V )