Theorem lpfval

Description: The limit point function on the subsets of a topology's base set. (Contributed by NM, 10-Feb-2007) (Revised by Mario Carneiro, 11-Nov-2013)

	Ref	Expression
Hypothesis	lpfval.1	⊢ 𝑋 = ∪ 𝐽
Assertion	lpfval	⊢ ( 𝐽 ∈ Top → ( limPt ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑦 ∣ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) } ) )

Proof

Step	Hyp	Ref	Expression
1		lpfval.1	⊢ 𝑋 = ∪ 𝐽
2	1	topopn	⊢ ( 𝐽 ∈ Top → 𝑋 ∈ 𝐽 )
3		pwexg	⊢ ( 𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V )
4		mptexg	⊢ ( 𝒫 𝑋 ∈ V → ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑦 ∣ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) } ) ∈ V )
5	2 3 4	3syl	⊢ ( 𝐽 ∈ Top → ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑦 ∣ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) } ) ∈ V )
6		unieq	⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽 )
7	6 1	eqtr4di	⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = 𝑋 )
8	7	pweqd	⊢ ( 𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋 )
9		fveq2	⊢ ( 𝑗 = 𝐽 → ( cls ‘ 𝑗 ) = ( cls ‘ 𝐽 ) )
10	9	fveq1d	⊢ ( 𝑗 = 𝐽 → ( ( cls ‘ 𝑗 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) = ( ( cls ‘ 𝐽 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) )
11	10	eleq2d	⊢ ( 𝑗 = 𝐽 → ( 𝑦 ∈ ( ( cls ‘ 𝑗 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) ↔ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) ) )
12	11	abbidv	⊢ ( 𝑗 = 𝐽 → { 𝑦 ∣ 𝑦 ∈ ( ( cls ‘ 𝑗 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) } = { 𝑦 ∣ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) } )
13	8 12	mpteq12dv	⊢ ( 𝑗 = 𝐽 → ( 𝑥 ∈ 𝒫 ∪ 𝑗 ↦ { 𝑦 ∣ 𝑦 ∈ ( ( cls ‘ 𝑗 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) } ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑦 ∣ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) } ) )
14		df-lp	⊢ limPt = ( 𝑗 ∈ Top ↦ ( 𝑥 ∈ 𝒫 ∪ 𝑗 ↦ { 𝑦 ∣ 𝑦 ∈ ( ( cls ‘ 𝑗 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) } ) )
15	13 14	fvmptg	⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑦 ∣ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) } ) ∈ V ) → ( limPt ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑦 ∣ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) } ) )
16	5 15	mpdan	⊢ ( 𝐽 ∈ Top → ( limPt ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑦 ∣ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑥 ∖ { 𝑦 } ) ) } ) )