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Theorem restperf 23116
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 = βˆͺ 𝐽
restcls.2 𝐾 = (𝐽 β†Ύt π‘Œ)
Assertion
Ref Expression
restperf ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ (𝐾 ∈ Perf ↔ π‘Œ βŠ† ((limPtβ€˜π½)β€˜π‘Œ)))
Proof of Theorem restperf
StepHypRef Expression
1 restcls.2 . . . . 5 𝐾 = (𝐽 β†Ύt π‘Œ)
2 restcls.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
32toptopon 22847 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
4 resttopon 23093 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
53, 4sylanb 579 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
61, 5eqeltrid 2833 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
7 topontop 22843 . . . 4 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
86, 7syl 17 . . 3 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ 𝐾 ∈ Top)
9 eqid 2728 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
109isperf 23083 . . . 4 (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾) = βˆͺ 𝐾))
1110baib 534 . . 3 (𝐾 ∈ Top β†’ (𝐾 ∈ Perf ↔ ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾) = βˆͺ 𝐾))
128, 11syl 17 . 2 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ (𝐾 ∈ Perf ↔ ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾) = βˆͺ 𝐾))
13 sseqin2 4217 . . 3 (π‘Œ βŠ† ((limPtβ€˜π½)β€˜π‘Œ) ↔ (((limPtβ€˜π½)β€˜π‘Œ) ∩ π‘Œ) = π‘Œ)
14 ssid 4004 . . . . . 6 π‘Œ βŠ† π‘Œ
152, 1restlp 23115 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ π‘Œ βŠ† π‘Œ) β†’ ((limPtβ€˜πΎ)β€˜π‘Œ) = (((limPtβ€˜π½)β€˜π‘Œ) ∩ π‘Œ))
1614, 15mp3an3 1446 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ ((limPtβ€˜πΎ)β€˜π‘Œ) = (((limPtβ€˜π½)β€˜π‘Œ) ∩ π‘Œ))
17 toponuni 22844 . . . . . . 7 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
186, 17syl 17 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ π‘Œ = βˆͺ 𝐾)
1918fveq2d 6906 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ ((limPtβ€˜πΎ)β€˜π‘Œ) = ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾))
2016, 19eqtr3d 2770 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ (((limPtβ€˜π½)β€˜π‘Œ) ∩ π‘Œ) = ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾))
2120, 18eqeq12d 2744 . . 3 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ ((((limPtβ€˜π½)β€˜π‘Œ) ∩ π‘Œ) = π‘Œ ↔ ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾) = βˆͺ 𝐾))
2213, 21bitrid 282 . 2 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ (π‘Œ βŠ† ((limPtβ€˜π½)β€˜π‘Œ) ↔ ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾) = βˆͺ 𝐾))
2312, 22bitr4d 281 1 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ (𝐾 ∈ Perf ↔ π‘Œ βŠ† ((limPtβ€˜π½)β€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ∩ cin 3948   βŠ† wss 3949  βˆͺ cuni 4912  β€˜cfv 6553  (class class class)co 7426   β†Ύt crest 17411  Topctop 22823  TopOnctopon 22840  limPtclp 23066  Perfcperf 23067
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-en 8973  df-fin 8976  df-fi 9444  df-rest 17413  df-topgen 17434  df-top 22824  df-topon 22841  df-bases 22877  df-cld 22951  df-cls 22953  df-lp 23068  df-perf 23069
This theorem is referenced by:  perfcls  23297  reperflem  24762  perfdvf  25860
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