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Theorem restlp 23117
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 = βˆͺ 𝐽
restcls.2 𝐾 = (𝐽 β†Ύt π‘Œ)
Assertion
Ref Expression
restlp ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ ((limPtβ€˜πΎ)β€˜π‘†) = (((limPtβ€˜π½)β€˜π‘†) ∩ π‘Œ))
Proof of Theorem restlp
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 simp3 1135 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝑆 βŠ† π‘Œ)
21ssdifssd 4140 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (𝑆 βˆ– {π‘₯}) βŠ† π‘Œ)
3 restcls.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
4 restcls.2 . . . . . . 7 𝐾 = (𝐽 β†Ύt π‘Œ)
53, 4restcls 23115 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ (𝑆 βˆ– {π‘₯}) βŠ† π‘Œ) β†’ ((clsβ€˜πΎ)β€˜(𝑆 βˆ– {π‘₯})) = (((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∩ π‘Œ))
62, 5syld3an3 1406 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ ((clsβ€˜πΎ)β€˜(𝑆 βˆ– {π‘₯})) = (((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∩ π‘Œ))
76eleq2d 2811 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (π‘₯ ∈ ((clsβ€˜πΎ)β€˜(𝑆 βˆ– {π‘₯})) ↔ π‘₯ ∈ (((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∩ π‘Œ)))
8 elin 3961 . . . 4 (π‘₯ ∈ (((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∩ π‘Œ) ↔ (π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∧ π‘₯ ∈ π‘Œ))
97, 8bitrdi 286 . . 3 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (π‘₯ ∈ ((clsβ€˜πΎ)β€˜(𝑆 βˆ– {π‘₯})) ↔ (π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∧ π‘₯ ∈ π‘Œ)))
10 simp1 1133 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝐽 ∈ Top)
113toptopon 22849 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
1210, 11sylib 217 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
13 simp2 1134 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ π‘Œ βŠ† 𝑋)
14 resttopon 23095 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
1512, 13, 14syl2anc 582 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
164, 15eqeltrid 2829 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
17 topontop 22845 . . . . 5 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
1816, 17syl 17 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝐾 ∈ Top)
19 toponuni 22846 . . . . . 6 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
2016, 19syl 17 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
211, 20sseqtrd 4018 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝑆 βŠ† βˆͺ 𝐾)
22 eqid 2725 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
2322islp 23074 . . . 4 ((𝐾 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐾) β†’ (π‘₯ ∈ ((limPtβ€˜πΎ)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜πΎ)β€˜(𝑆 βˆ– {π‘₯}))))
2418, 21, 23syl2anc 582 . . 3 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (π‘₯ ∈ ((limPtβ€˜πΎ)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜πΎ)β€˜(𝑆 βˆ– {π‘₯}))))
25 elin 3961 . . . 4 (π‘₯ ∈ (((limPtβ€˜π½)β€˜π‘†) ∩ π‘Œ) ↔ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ∧ π‘₯ ∈ π‘Œ))
261, 13sstrd 3988 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝑆 βŠ† 𝑋)
273islp 23074 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
2810, 26, 27syl2anc 582 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
2928anbi1d 629 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ ((π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ∧ π‘₯ ∈ π‘Œ) ↔ (π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∧ π‘₯ ∈ π‘Œ)))
3025, 29bitrid 282 . . 3 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (π‘₯ ∈ (((limPtβ€˜π½)β€˜π‘†) ∩ π‘Œ) ↔ (π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∧ π‘₯ ∈ π‘Œ)))
319, 24, 303bitr4d 310 . 2 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (π‘₯ ∈ ((limPtβ€˜πΎ)β€˜π‘†) ↔ π‘₯ ∈ (((limPtβ€˜π½)β€˜π‘†) ∩ π‘Œ)))
3231eqrdv 2723 1 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ ((limPtβ€˜πΎ)β€˜π‘†) = (((limPtβ€˜π½)β€˜π‘†) ∩ π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βˆ– cdif 3942   ∩ cin 3944   βŠ† wss 3945  {csn 4629  βˆͺ cuni 4908  β€˜cfv 6547  (class class class)co 7417   β†Ύt crest 17401  Topctop 22825  TopOnctopon 22842  clsccl 22952  limPtclp 23068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7870  df-1st 7992  df-2nd 7993  df-en 8963  df-fin 8966  df-fi 9434  df-rest 17403  df-topgen 17424  df-top 22826  df-topon 22843  df-bases 22879  df-cld 22953  df-cls 22955  df-lp 23070
This theorem is referenced by:  restperf  23118  lptioo2cn  45096  lptioo1cn  45097  limclner  45102  fourierdlem42  45600
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