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Theorem restlp 23207
Description: The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
restcls.1 𝑋 = 𝐽
restcls.2 𝐾 = (𝐽t 𝑌)
Assertion
Ref Expression
restlp ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌))

Proof of Theorem restlp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp3 1137 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑌)
21ssdifssd 4157 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑆 ∖ {𝑥}) ⊆ 𝑌)
3 restcls.1 . . . . . . 7 𝑋 = 𝐽
4 restcls.2 . . . . . . 7 𝐾 = (𝐽t 𝑌)
53, 4restcls 23205 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑌) → ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) = (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌))
62, 5syld3an3 1408 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) = (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌))
76eleq2d 2825 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) ↔ 𝑥 ∈ (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌)))
8 elin 3979 . . . 4 (𝑥 ∈ (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥𝑌))
97, 8bitrdi 287 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥𝑌)))
10 simp1 1135 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ Top)
113toptopon 22939 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
1210, 11sylib 218 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ (TopOn‘𝑋))
13 simp2 1136 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌𝑋)
14 resttopon 23185 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
1512, 13, 14syl2anc 584 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
164, 15eqeltrid 2843 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ (TopOn‘𝑌))
17 topontop 22935 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
1816, 17syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ Top)
19 toponuni 22936 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
2016, 19syl 17 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 = 𝐾)
211, 20sseqtrd 4036 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 𝐾)
22 eqid 2735 . . . . 5 𝐾 = 𝐾
2322islp 23164 . . . 4 ((𝐾 ∈ Top ∧ 𝑆 𝐾) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥}))))
2418, 21, 23syl2anc 584 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥}))))
25 elin 3979 . . . 4 (𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌) ↔ (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ∧ 𝑥𝑌))
261, 13sstrd 4006 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑋)
273islp 23164 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
2810, 26, 27syl2anc 584 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
2928anbi1d 631 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((𝑥 ∈ ((limPt‘𝐽)‘𝑆) ∧ 𝑥𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥𝑌)))
3025, 29bitrid 283 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥𝑌)))
319, 24, 303bitr4d 311 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌)))
3231eqrdv 2733 1 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cdif 3960  cin 3962  wss 3963  {csn 4631   cuni 4912  cfv 6563  (class class class)co 7431  t crest 17467  Topctop 22915  TopOnctopon 22932  clsccl 23042  limPtclp 23158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-cls 23045  df-lp 23160
This theorem is referenced by:  restperf  23208  lptioo2cn  45601  lptioo1cn  45602  limclner  45607  fourierdlem42  46105
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