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Theorem restperf 20798
 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 20548 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4 resttopon 20775 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
53, 4sylanb 488 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
61, 5syl5eqel 2692 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝐾 ∈ (TopOn‘𝑌))
7 topontop 20541 . . . 4 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
86, 7syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝐾 ∈ Top)
9 eqid 2610 . . . . 5 𝐾 = 𝐾
109isperf 20765 . . . 4 (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
1110baib 942 . . 3 (𝐾 ∈ Top → (𝐾 ∈ Perf ↔ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
128, 11syl 17 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐾 ∈ Perf ↔ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
13 sseqin2 3779 . . 3 (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌)
14 ssid 3587 . . . . . 6 𝑌𝑌
152, 1restlp 20797 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑌𝑌) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌))
1614, 15mp3an3 1405 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌))
17 toponuni 20542 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
186, 17syl 17 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 = 𝐾)
1918fveq2d 6107 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((limPt‘𝐾)‘𝑌) = ((limPt‘𝐾)‘ 𝐾))
2016, 19eqtr3d 2646 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = ((limPt‘𝐾)‘ 𝐾))
2120, 18eqeq12d 2625 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌 ↔ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
2213, 21syl5bb 271 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
2312, 22bitr4d 270 1 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  TopOnctopon 20518  limPtclp 20748  Perfcperf 20749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-cls 20635  df-lp 20750  df-perf 20751 This theorem is referenced by:  perfcls  20979  reperflem  22429  perfdvf  23473
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