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Theorem lpval 21722
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 21721 . . . 4 (𝐽 ∈ Top → (limPt‘𝐽) = (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))}))
32fveq1d 6645 . . 3 (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆))
43adantr 484 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆))
5 eqid 2821 . . 3 (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))}) = (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})
6 difeq1 4068 . . . . . 6 (𝑦 = 𝑆 → (𝑦 ∖ {𝑥}) = (𝑆 ∖ {𝑥}))
76fveq2d 6647 . . . . 5 (𝑦 = 𝑆 → ((cls‘𝐽)‘(𝑦 ∖ {𝑥})) = ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
87eleq2d 2897 . . . 4 (𝑦 = 𝑆 → (𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥})) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
98abbidv 2885 . . 3 (𝑦 = 𝑆 → {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))} = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
101topopn 21489 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
11 elpw2g 5220 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1210, 11syl 17 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1312biimpar 481 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
1410adantr 484 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋𝐽)
15 ssdifss 4088 . . . . . 6 (𝑆𝑋 → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
161clsss3 21642 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ 𝑋)
1716sseld 3942 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥𝑋))
1815, 17sylan2 595 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥𝑋))
1918abssdv 4021 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ⊆ 𝑋)
2014, 19ssexd 5201 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ∈ V)
215, 9, 13, 20fvmptd3 6764 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆) = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
224, 21eqtrd 2856 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {cab 2799  Vcvv 3471  cdif 3907  wss 3910  𝒫 cpw 4512  {csn 4540   cuni 4811  cmpt 5119  cfv 6328  Topctop 21476  clsccl 21601  limPtclp 21717
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-top 21477  df-cld 21602  df-cls 21604  df-lp 21719
This theorem is referenced by:  islp  21723  lpsscls  21724
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