Theorem islp

Description: The predicate "the class P is a limit point of S ". (Contributed by NM, 10-Feb-2007)

	Ref	Expression
Hypothesis	lpfval.1	⊢ 𝑋 = ∪ 𝐽
Assertion	islp	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑃 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) ↔ 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		lpfval.1	⊢ 𝑋 = ∪ 𝐽
2	1	lpval	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) = { 𝑥 ∣ 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) } )
3	2	eleq2d	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑃 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) ↔ 𝑃 ∈ { 𝑥 ∣ 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) } ) )
4		id	⊢ ( 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) → 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) )
5		id	⊢ ( 𝑥 = 𝑃 → 𝑥 = 𝑃 )
6		sneq	⊢ ( 𝑥 = 𝑃 → { 𝑥 } = { 𝑃 } )
7	6	difeq2d	⊢ ( 𝑥 = 𝑃 → ( 𝑆 ∖ { 𝑥 } ) = ( 𝑆 ∖ { 𝑃 } ) )
8	7	fveq2d	⊢ ( 𝑥 = 𝑃 → ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) = ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) )
9	5 8	eleq12d	⊢ ( 𝑥 = 𝑃 → ( 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) ↔ 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) ) )
10	4 9	elab3	⊢ ( 𝑃 ∈ { 𝑥 ∣ 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) } ↔ 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) )
11	3 10	bitrdi	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑃 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) ↔ 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) ) )