Theorem kur14lem5

Description: Lemma for kur14 . Closure is an idempotent operation in the set of subsets of a topology. (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	kur14lem5	|- ( K ` ( K ` A ) ) = ( K ` A )

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	2	clsidm	|- ( ( J e. Top /\ A C_ X ) -> ( ( cls ` J ) ` ( ( cls ` J ) ` A ) ) = ( ( cls ` J ) ` A ) )
7	1 5 6	mp2an	|- ( ( cls ` J ) ` ( ( cls ` J ) ` A ) ) = ( ( cls ` J ) ` A )
8	3	fveq1i	|- ( K ` A ) = ( ( cls ` J ) ` A )
9	3 8	fveq12i	|- ( K ` ( K ` A ) ) = ( ( cls ` J ) ` ( ( cls ` J ) ` A ) )
10	7 9 8	3eqtr4i	|- ( K ` ( K ` A ) ) = ( K ` A )