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Theorem restperf 21798
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 21531 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4 resttopon 21775 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
53, 4sylanb 584 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
61, 5eqeltrid 2920 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝐾 ∈ (TopOn‘𝑌))
7 topontop 21527 . . . 4 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
86, 7syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝐾 ∈ Top)
9 eqid 2824 . . . . 5 𝐾 = 𝐾
109isperf 21765 . . . 4 (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
1110baib 539 . . 3 (𝐾 ∈ Top → (𝐾 ∈ Perf ↔ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
128, 11syl 17 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐾 ∈ Perf ↔ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
13 sseqin2 4177 . . 3 (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌)
14 ssid 3975 . . . . . 6 𝑌𝑌
152, 1restlp 21797 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑌𝑌) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌))
1614, 15mp3an3 1447 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌))
17 toponuni 21528 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
186, 17syl 17 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 = 𝐾)
1918fveq2d 6667 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((limPt‘𝐾)‘𝑌) = ((limPt‘𝐾)‘ 𝐾))
2016, 19eqtr3d 2861 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = ((limPt‘𝐾)‘ 𝐾))
2120, 18eqeq12d 2840 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌 ↔ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
2213, 21syl5bb 286 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ ((limPt‘𝐾)‘ 𝐾) = 𝐾))
2312, 22bitr4d 285 1 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cin 3918  wss 3919   cuni 4824  cfv 6345  (class class class)co 7151  t crest 16696  Topctop 21507  TopOnctopon 21524  limPtclp 21748  Perfcperf 21749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-oadd 8104  df-er 8287  df-en 8508  df-fin 8511  df-fi 8874  df-rest 16698  df-topgen 16719  df-top 21508  df-topon 21525  df-bases 21560  df-cld 21633  df-cls 21635  df-lp 21750  df-perf 21751
This theorem is referenced by:  perfcls  21979  reperflem  23432  perfdvf  24515
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