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Theorem lpval 20924
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 20923 . . . 4 (𝐽 ∈ Top → (limPt‘𝐽) = (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))}))
32fveq1d 6180 . . 3 (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆))
43adantr 481 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆))
51topopn 20692 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
6 elpw2g 4818 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
75, 6syl 17 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
87biimpar 502 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
95adantr 481 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋𝐽)
10 ssdifss 3733 . . . . . 6 (𝑆𝑋 → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
111clsss3 20844 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ 𝑋)
1211sseld 3594 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥𝑋))
1310, 12sylan2 491 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥𝑋))
1413abssdv 3668 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ⊆ 𝑋)
159, 14ssexd 4796 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ∈ V)
16 difeq1 3713 . . . . . . 7 (𝑦 = 𝑆 → (𝑦 ∖ {𝑥}) = (𝑆 ∖ {𝑥}))
1716fveq2d 6182 . . . . . 6 (𝑦 = 𝑆 → ((cls‘𝐽)‘(𝑦 ∖ {𝑥})) = ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
1817eleq2d 2685 . . . . 5 (𝑦 = 𝑆 → (𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥})) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
1918abbidv 2739 . . . 4 (𝑦 = 𝑆 → {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))} = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
20 eqid 2620 . . . 4 (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))}) = (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})
2119, 20fvmptg 6267 . . 3 ((𝑆 ∈ 𝒫 𝑋 ∧ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ∈ V) → ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆) = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
228, 15, 21syl2anc 692 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆) = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
234, 22eqtrd 2654 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  {cab 2606  Vcvv 3195  cdif 3564  wss 3567  𝒫 cpw 4149  {csn 4168   cuni 4427  cmpt 4720  cfv 5876  Topctop 20679  clsccl 20803  limPtclp 20919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-top 20680  df-cld 20804  df-cls 20806  df-lp 20921
This theorem is referenced by:  islp  20925  lpsscls  20926
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