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 ∈ Top
	kur14lem.x	⊢ 𝑋 = ∪ 𝐽
	kur14lem.k	⊢ 𝐾 = ( cls ‘ 𝐽 )
	kur14lem.i	⊢ 𝐼 = ( int ‘ 𝐽 )
	kur14lem.a	⊢ 𝐴 ⊆ 𝑋
Assertion	kur14lem3	⊢ ( 𝐾 ‘ 𝐴 ) ⊆ 𝑋

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	3	fveq1i	⊢ ( 𝐾 ‘ 𝐴 ) = ( ( cls ‘ 𝐽 ) ‘ 𝐴 )
7	2	clsss3	⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ⊆ 𝑋 )
8	1 5 7	mp2an	⊢ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ⊆ 𝑋
9	6 8	eqsstri	⊢ ( 𝐾 ‘ 𝐴 ) ⊆ 𝑋