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 ∈ Top
	kur14lem.x	⊢ 𝑋 = ∪ 𝐽
	kur14lem.k	⊢ 𝐾 = ( cls ‘ 𝐽 )
	kur14lem.i	⊢ 𝐼 = ( int ‘ 𝐽 )
	kur14lem.a	⊢ 𝐴 ⊆ 𝑋
Assertion	kur14lem5	⊢ ( 𝐾 ‘ ( 𝐾 ‘ 𝐴 ) ) = ( 𝐾 ‘ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		kur14lem.j	⊢ 𝐽 ∈ Top
2		kur14lem.x	⊢ 𝑋 = ∪ 𝐽
3		kur14lem.k	⊢ 𝐾 = ( cls ‘ 𝐽 )
4		kur14lem.i	⊢ 𝐼 = ( int ‘ 𝐽 )
5		kur14lem.a	⊢ 𝐴 ⊆ 𝑋
6	2	clsidm	⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 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 )