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Theorem lpval 23147
Description: The set of limit points of a subset of the base set of a topology. Alternate definition of limit point in [Munkres] p. 97. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = 𝐽
Assertion
Ref Expression
lpval ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
Distinct variable groups:   𝑥,𝐽   𝑥,𝑆   𝑥,𝑋

Proof of Theorem lpval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 lpfval.1 . . . . 5 𝑋 = 𝐽
21lpfval 23146 . . . 4 (𝐽 ∈ Top → (limPt‘𝐽) = (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))}))
32fveq1d 6908 . . 3 (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆))
43adantr 480 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆))
5 eqid 2737 . . 3 (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))}) = (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})
6 difeq1 4119 . . . . . 6 (𝑦 = 𝑆 → (𝑦 ∖ {𝑥}) = (𝑆 ∖ {𝑥}))
76fveq2d 6910 . . . . 5 (𝑦 = 𝑆 → ((cls‘𝐽)‘(𝑦 ∖ {𝑥})) = ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
87eleq2d 2827 . . . 4 (𝑦 = 𝑆 → (𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥})) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
98abbidv 2808 . . 3 (𝑦 = 𝑆 → {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))} = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
101topopn 22912 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
11 elpw2g 5333 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1210, 11syl 17 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1312biimpar 477 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
1410adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋𝐽)
15 ssdifss 4140 . . . . . 6 (𝑆𝑋 → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
161clsss3 23067 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ 𝑋)
1716sseld 3982 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥𝑋))
1815, 17sylan2 593 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥𝑋))
1918abssdv 4068 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ⊆ 𝑋)
2014, 19ssexd 5324 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ∈ V)
215, 9, 13, 20fvmptd3 7039 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆) = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
224, 21eqtrd 2777 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  cdif 3948  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907  cmpt 5225  cfv 6561  Topctop 22899  clsccl 23026  limPtclp 23142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-cls 23029  df-lp 23144
This theorem is referenced by:  islp  23148  lpsscls  23149
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