MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lpfval Structured version   Visualization version   GIF version

Theorem lpfval 22993
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 22759 . . 3 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
3 pwexg 5369 . . 3 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
4 mptexg 7217 . . 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 4913 . . . . . 6 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
76, 1eqtr4di 2784 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = 𝑋)
87pweqd 4614 . . . 4 (𝑗 = 𝐽 β†’ 𝒫 βˆͺ 𝑗 = 𝒫 𝑋)
9 fveq2 6884 . . . . . . 7 (𝑗 = 𝐽 β†’ (clsβ€˜π‘—) = (clsβ€˜π½))
109fveq1d 6886 . . . . . 6 (𝑗 = 𝐽 β†’ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦})) = ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦})))
1110eleq2d 2813 . . . . 5 (𝑗 = 𝐽 β†’ (𝑦 ∈ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦})) ↔ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))))
1211abbidv 2795 . . . 4 (𝑗 = 𝐽 β†’ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦}))} = {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π½)β€˜(π‘₯ βˆ– {𝑦}))})
138, 12mpteq12dv 5232 . . 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 22991 . . 3 limPt = (𝑗 ∈ Top ↦ (π‘₯ ∈ 𝒫 βˆͺ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((clsβ€˜π‘—)β€˜(π‘₯ βˆ– {𝑦}))}))
1513, 14fvmptg 6989 . 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 684 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 2703  Vcvv 3468   βˆ– cdif 3940  π’« cpw 4597  {csn 4623  βˆͺ cuni 4902   ↦ cmpt 5224  β€˜cfv 6536  Topctop 22746  clsccl 22873  limPtclp 22989
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 22747  df-lp 22991
This theorem is referenced by:  lpval  22994
  Copyright terms: Public domain W3C validator