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Theorem lpfval 22473
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 = βˆͺ 𝐽
Assertion
Ref Expression
lpfval (𝐽 ∈ Top β†’ (limPtβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))}))
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝑋,𝑦

Proof of Theorem lpfval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 lpfval.1 . . . 4 𝑋 = βˆͺ 𝐽
21topopn 22239 . . 3 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
3 pwexg 5331 . . 3 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
4 mptexg 7167 . . 3 (𝒫 𝑋 ∈ V β†’ (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))}) ∈ V)
52, 3, 43syl 18 . 2 (𝐽 ∈ Top β†’ (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))}) ∈ V)
6 unieq 4874 . . . . . 6 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
76, 1eqtr4di 2794 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = 𝑋)
87pweqd 4575 . . . 4 (𝑗 = 𝐽 β†’ 𝒫 βˆͺ 𝑗 = 𝒫 𝑋)
9 fveq2 6839 . . . . . . 7 (𝑗 = 𝐽 β†’ (clsβ€˜π‘—) = (clsβ€˜π½))
109fveq1d 6841 . . . . . 6 (𝑗 = 𝐽 β†’ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦})) = ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦})))
1110eleq2d 2823 . . . . 5 (𝑗 = 𝐽 β†’ (𝑦 ∈ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦})) ↔ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))))
1211abbidv 2805 . . . 4 (𝑗 = 𝐽 β†’ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦}))} = {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))})
138, 12mpteq12dv 5194 . . 3 (𝑗 = 𝐽 β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦}))}) = (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))}))
14 df-lp 22471 . . 3 limPt = (𝑗 ∈ Top ↦ (π‘₯ ∈ 𝒫 βˆͺ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦}))}))
1513, 14fvmptg 6943 . 2 ((𝐽 ∈ Top ∧ (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))}) ∈ V) β†’ (limPtβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))}))
165, 15mpdan 685 1 (𝐽 ∈ Top β†’ (limPtβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  {cab 2713  Vcvv 3443   βˆ– cdif 3905  π’« cpw 4558  {csn 4584  βˆͺ cuni 4863   ↦ cmpt 5186  β€˜cfv 6493  Topctop 22226  clsccl 22353  limPtclp 22469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-top 22227  df-lp 22471
This theorem is referenced by:  lpval  22474
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