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Theorem lpfval 20852
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.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = 𝐽
Assertion
Ref Expression
lpfval (𝐽 ∈ Top → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦

Proof of Theorem lpfval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 lpfval.1 . . . 4 𝑋 = 𝐽
21topopn 20636 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 4810 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 mptexg 6438 . . 3 (𝒫 𝑋 ∈ V → (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}) ∈ V)
52, 3, 43syl 18 . 2 (𝐽 ∈ Top → (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}) ∈ V)
6 unieq 4410 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1syl6eqr 2673 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 4135 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
9 fveq2 6148 . . . . . . 7 (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
109fveq1d 6150 . . . . . 6 (𝑗 = 𝐽 → ((cls‘𝑗)‘(𝑥 ∖ {𝑦})) = ((cls‘𝐽)‘(𝑥 ∖ {𝑦})))
1110eleq2d 2684 . . . . 5 (𝑗 = 𝐽 → (𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦})) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))))
1211abbidv 2738 . . . 4 (𝑗 = 𝐽 → {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))} = {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))})
138, 12mpteq12dv 4693 . . 3 (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
14 df-lp 20850 . . 3 limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
1513, 14fvmptg 6237 . 2 ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}) ∈ V) → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
165, 15mpdan 701 1 (𝐽 ∈ Top → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3186  cdif 3552  𝒫 cpw 4130  {csn 4148   cuni 4402  cmpt 4673  cfv 5847  Topctop 20617  clsccl 20732  limPtclp 20848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-top 20621  df-lp 20850
This theorem is referenced by:  lpval  20853
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