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

Theorem islp 23163
Description: The predicate "the class 𝑃 is a limit point of 𝑆". (Contributed by NM, 10-Feb-2007.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = 𝐽
Assertion
Ref Expression
islp ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))

Proof of Theorem islp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 lpfval.1 . . . 4 𝑋 = 𝐽
21lpval 23162 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
32eleq2d 2824 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))}))
4 id 22 . . 3 (𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})) → 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})))
5 id 22 . . . 4 (𝑥 = 𝑃𝑥 = 𝑃)
6 sneq 4640 . . . . . 6 (𝑥 = 𝑃 → {𝑥} = {𝑃})
76difeq2d 4135 . . . . 5 (𝑥 = 𝑃 → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑃}))
87fveq2d 6910 . . . 4 (𝑥 = 𝑃 → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) = ((cls‘𝐽)‘(𝑆 ∖ {𝑃})))
95, 8eleq12d 2832 . . 3 (𝑥 = 𝑃 → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))
104, 9elab3 3688 . 2 (𝑃 ∈ {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})))
113, 10bitrdi 287 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  {cab 2711  cdif 3959  wss 3962  {csn 4630   cuni 4911  cfv 6562  Topctop 22914  clsccl 23041  limPtclp 23157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-top 22915  df-cld 23042  df-cls 23044  df-lp 23159
This theorem is referenced by:  lpdifsn  23166  lpss3  23167  islp2  23168  islp3  23169  maxlp  23170  restlp  23206  lpcls  23387  limcnlp  25927  limcflflem  25929
  Copyright terms: Public domain W3C validator