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Theorem restperf 21486
Description: Perfection of a subspace. Note that the term "perfect set" is reserved for closed sets which are perfect in the subspace topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
restcls.1 𝑋 = 𝐽
restcls.2 𝐾 = (𝐽t 𝑌)
Assertion
Ref Expression
restperf ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌)))

Proof of Theorem restperf
StepHypRef Expression
1 restcls.2 . . . . 5 𝐾 = (𝐽t 𝑌)
2 restcls.1 . . . . . . 7 𝑋 = 𝐽
32toptopon 21219 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4 resttopon 21463 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
53, 4sylanb 573 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
61, 5syl5eqel 2864 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝐾 ∈ (TopOn‘𝑌))
7 topontop 21215 . . . 4 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
86, 7syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝐾 ∈ Top)
9 eqid 2772 . . . . 5 𝐾 = 𝐾
109isperf 21453 . . . 4 (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
1110baib 528 . . 3 (𝐾 ∈ Top → (𝐾 ∈ Perf ↔ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
128, 11syl 17 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐾 ∈ Perf ↔ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
13 sseqin2 4074 . . 3 (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌)
14 ssid 3875 . . . . . 6 𝑌𝑌
152, 1restlp 21485 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑌𝑌) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌))
1614, 15mp3an3 1429 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌))
17 toponuni 21216 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
186, 17syl 17 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 = 𝐾)
1918fveq2d 6497 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((limPt‘𝐾)‘𝑌) = ((limPt‘𝐾)‘ 𝐾))
2016, 19eqtr3d 2810 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = ((limPt‘𝐾)‘ 𝐾))
2120, 18eqeq12d 2787 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌 ↔ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
2213, 21syl5bb 275 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
2312, 22bitr4d 274 1 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2048  cin 3824  wss 3825   cuni 4706  cfv 6182  (class class class)co 6970  t crest 16540  Topctop 21195  TopOnctopon 21212  limPtclp 21436  Perfcperf 21437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-oadd 7901  df-er 8081  df-en 8299  df-fin 8302  df-fi 8662  df-rest 16542  df-topgen 16563  df-top 21196  df-topon 21213  df-bases 21248  df-cld 21321  df-cls 21323  df-lp 21438  df-perf 21439
This theorem is referenced by:  perfcls  21667  reperflem  23119  perfdvf  24194
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