Theorem clsss2

Description: If a subset is included in a closed set, so is the subset's closure. (Contributed by NM, 22-Feb-2007)

	Ref	Expression
Hypothesis	clscld.1	|- X = U. J
Assertion	clsss2	|- ( ( C e. ( Clsd ` J ) /\ S C_ C ) -> ( ( cls ` J ) ` S ) C_ C )

Proof

Step	Hyp	Ref	Expression
1		clscld.1	|- X = U. J
2		cldrcl	|- ( C e. ( Clsd ` J ) -> J e. Top )
3	2	adantr	|- ( ( C e. ( Clsd ` J ) /\ S C_ C ) -> J e. Top )
4	1	cldss	|- ( C e. ( Clsd ` J ) -> C C_ X )
5	4	adantr	|- ( ( C e. ( Clsd ` J ) /\ S C_ C ) -> C C_ X )
6		simpr	|- ( ( C e. ( Clsd ` J ) /\ S C_ C ) -> S C_ C )
7	1	clsss	|- ( ( J e. Top /\ C C_ X /\ S C_ C ) -> ( ( cls ` J ) ` S ) C_ ( ( cls ` J ) ` C ) )
8	3 5 6 7	syl3anc	|- ( ( C e. ( Clsd ` J ) /\ S C_ C ) -> ( ( cls ` J ) ` S ) C_ ( ( cls ` J ) ` C ) )
9		cldcls	|- ( C e. ( Clsd ` J ) -> ( ( cls ` J ) ` C ) = C )
10	9	adantr	|- ( ( C e. ( Clsd ` J ) /\ S C_ C ) -> ( ( cls ` J ) ` C ) = C )
11	8 10	sseqtrd	|- ( ( C e. ( Clsd ` J ) /\ S C_ C ) -> ( ( cls ` J ) ` S ) C_ C )