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Theorem islp 22419
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 22418 . . 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 4595 . . . . . 6 (𝑥 = 𝑃 → {𝑥} = {𝑃})
76difeq2d 4081 . . . . 5 (𝑥 = 𝑃 → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑃}))
87fveq2d 6842 . . . 4 (𝑥 = 𝑃 → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) = ((cls‘𝐽)‘(𝑆 ∖ {𝑃})))
95, 8eleq12d 2833 . . 3 (𝑥 = 𝑃 → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))
104, 9elab3 3637 . 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 205  wa 397   = wceq 1542  wcel 2107  {cab 2715  cdif 3906  wss 3909  {csn 4585   cuni 4864  cfv 6492  Topctop 22170  clsccl 22297  limPtclp 22413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-iin 4956  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-top 22171  df-cld 22298  df-cls 22300  df-lp 22415
This theorem is referenced by:  lpdifsn  22422  lpss3  22423  islp2  22424  islp3  22425  maxlp  22426  restlp  22462  lpcls  22643  limcnlp  25170  limcflflem  25172
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