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Metamath Proof Explorer

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 } ) ) } ) )