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Theorem islp2 21169
Description: The predicate "𝑃 is a limit point of 𝑆," in terms of neighborhoods. Definition of limit point in [Munkres] p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods. (Contributed by NM, 26-Feb-2007.) (Proof shortened by Mario Carneiro, 25-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = 𝐽
Assertion
Ref Expression
islp2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
Distinct variable groups:   𝑛,𝐽   𝑃,𝑛   𝑆,𝑛   𝑛,𝑋

Proof of Theorem islp2
StepHypRef Expression
1 lpfval.1 . . . 4 𝑋 = 𝐽
21islp 21164 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))
323adant3 1125 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))
4 ssdifss 3890 . . 3 (𝑆𝑋 → (𝑆 ∖ {𝑃}) ⊆ 𝑋)
51neindisj2 21147 . . 3 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑃}) ⊆ 𝑋𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
64, 5syl3an2 1166 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
73, 6bitrd 268 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  cdif 3718  cin 3720  wss 3721  c0 4061  {csn 4314   cuni 4572  cfv 6031  Topctop 20917  clsccl 21042  neicnei 21121  limPtclp 21158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-top 20918  df-cld 21043  df-ntr 21044  df-cls 21045  df-nei 21122  df-lp 21160
This theorem is referenced by:  clslp  21172  lpbl  22527  reperflem  22840  islptre  40363  islpcn  40383
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