Theorem kur14lem3

Description: Lemma for kur14 . A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015)

	Ref	Expression
Hypotheses	kur14lem.j	|- J e. Top
	kur14lem.x	|- X = U. J
	kur14lem.k	|- K = ( cls ` J )
	kur14lem.i	|- I = ( int ` J )
	kur14lem.a	|- A C_ X
Assertion	kur14lem3	|- ( K ` A ) C_ X

Proof

Step	Hyp	Ref	Expression
1		kur14lem.j	|- J e. Top
2		kur14lem.x	|- X = U. J
3		kur14lem.k	|- K = ( cls ` J )
4		kur14lem.i	|- I = ( int ` J )
5		kur14lem.a	|- A C_ X
6	3	fveq1i	|- ( K ` A ) = ( ( cls ` J ) ` A )
7	2	clsss3	|- ( ( J e. Top /\ A C_ X ) -> ( ( cls ` J ) ` A ) C_ X )
8	1 5 7	mp2an	|- ( ( cls ` J ) ` A ) C_ X
9	6 8	eqsstri	|- ( K ` A ) C_ X