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Theorem lpfval 23060
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 22826 . . 3 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
3 pwexg 5380 . . 3 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
4 mptexg 7237 . . 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 4921 . . . . . 6 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
76, 1eqtr4di 2785 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = 𝑋)
87pweqd 4621 . . . 4 (𝑗 = 𝐽 β†’ 𝒫 βˆͺ 𝑗 = 𝒫 𝑋)
9 fveq2 6900 . . . . . . 7 (𝑗 = 𝐽 β†’ (clsβ€˜π‘—) = (clsβ€˜π½))
109fveq1d 6902 . . . . . 6 (𝑗 = 𝐽 β†’ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦})) = ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦})))
1110eleq2d 2814 . . . . 5 (𝑗 = 𝐽 β†’ (𝑦 ∈ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦})) ↔ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))))
1211abbidv 2796 . . . 4 (𝑗 = 𝐽 β†’ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦}))} = {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))})
138, 12mpteq12dv 5241 . . 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 23058 . . 3 limPt = (𝑗 ∈ Top ↦ (π‘₯ ∈ 𝒫 βˆͺ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦}))}))
1513, 14fvmptg 7006 . 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 1533   ∈ wcel 2098  {cab 2704  Vcvv 3471   βˆ– cdif 3944  π’« cpw 4604  {csn 4630  βˆͺ cuni 4910   ↦ cmpt 5233  β€˜cfv 6551  Topctop 22813  clsccl 22940  limPtclp 23056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-top 22814  df-lp 23058
This theorem is referenced by:  lpval  23061
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