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Theorem lpfval 21313
 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 21081 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 5078 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 mptexg 6740 . . 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 4666 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1syl6eqr 2879 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 4383 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
9 fveq2 6433 . . . . . . 7 (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
109fveq1d 6435 . . . . . 6 (𝑗 = 𝐽 → ((cls‘𝑗)‘(𝑥 ∖ {𝑦})) = ((cls‘𝐽)‘(𝑥 ∖ {𝑦})))
1110eleq2d 2892 . . . . 5 (𝑗 = 𝐽 → (𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦})) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))))
1211abbidv 2946 . . . 4 (𝑗 = 𝐽 → {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))} = {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))})
138, 12mpteq12dv 4956 . . 3 (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
14 df-lp 21311 . . 3 limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
1513, 14fvmptg 6527 . 2 ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}) ∈ V) → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
165, 15mpdan 680 1 (𝐽 ∈ Top → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  {cab 2811  Vcvv 3414   ∖ cdif 3795  𝒫 cpw 4378  {csn 4397  ∪ cuni 4658   ↦ cmpt 4952  ‘cfv 6123  Topctop 21068  clsccl 21193  limPtclp 21309 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-top 21069  df-lp 21311 This theorem is referenced by:  lpval  21314
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