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Theorem kgencmp2 22082
Description: The compact generator topology has the same compact sets as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
kgencmp2 (𝐽 ∈ Top → ((𝐽t 𝐾) ∈ Comp ↔ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp))

Proof of Theorem kgencmp2
StepHypRef Expression
1 kgencmp 22081 . . 3 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) = ((𝑘Gen‘𝐽) ↾t 𝐾))
2 simpr 485 . . 3 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Comp)
31, 2eqeltrrd 2911 . 2 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp)
4 cmptop 21931 . . . . . . 7 (((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp → ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Top)
5 restrcl 21693 . . . . . . . 8 (((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Top → ((𝑘Gen‘𝐽) ∈ V ∧ 𝐾 ∈ V))
65simprd 496 . . . . . . 7 (((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Top → 𝐾 ∈ V)
74, 6syl 17 . . . . . 6 (((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp → 𝐾 ∈ V)
8 resttop 21696 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐾 ∈ V) → (𝐽t 𝐾) ∈ Top)
97, 8sylan2 592 . . . . 5 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Top)
10 toptopon2 21454 . . . . 5 ((𝐽t 𝐾) ∈ Top ↔ (𝐽t 𝐾) ∈ (TopOn‘ (𝐽t 𝐾)))
119, 10sylib 219 . . . 4 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ (TopOn‘ (𝐽t 𝐾)))
12 eqid 2818 . . . . . . . . 9 𝐽 = 𝐽
1312kgenuni 22075 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 = (𝑘Gen‘𝐽))
1413adantr 481 . . . . . . 7 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → 𝐽 = (𝑘Gen‘𝐽))
1514ineq2d 4186 . . . . . 6 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → (𝐾 𝐽) = (𝐾 (𝑘Gen‘𝐽)))
1612restuni2 21703 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐾 ∈ V) → (𝐾 𝐽) = (𝐽t 𝐾))
177, 16sylan2 592 . . . . . 6 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → (𝐾 𝐽) = (𝐽t 𝐾))
18 kgenftop 22076 . . . . . . 7 (𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ Top)
19 eqid 2818 . . . . . . . 8 (𝑘Gen‘𝐽) = (𝑘Gen‘𝐽)
2019restuni2 21703 . . . . . . 7 (((𝑘Gen‘𝐽) ∈ Top ∧ 𝐾 ∈ V) → (𝐾 (𝑘Gen‘𝐽)) = ((𝑘Gen‘𝐽) ↾t 𝐾))
2118, 7, 20syl2an 595 . . . . . 6 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → (𝐾 (𝑘Gen‘𝐽)) = ((𝑘Gen‘𝐽) ↾t 𝐾))
2215, 17, 213eqtr3d 2861 . . . . 5 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → (𝐽t 𝐾) = ((𝑘Gen‘𝐽) ↾t 𝐾))
2322fveq2d 6667 . . . 4 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → (TopOn‘ (𝐽t 𝐾)) = (TopOn‘ ((𝑘Gen‘𝐽) ↾t 𝐾)))
2411, 23eleqtrd 2912 . . 3 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ (TopOn‘ ((𝑘Gen‘𝐽) ↾t 𝐾)))
25 simpr 485 . . 3 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp)
26 kgenss 22079 . . . . 5 (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
2726adantr 481 . . . 4 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → 𝐽 ⊆ (𝑘Gen‘𝐽))
28 ssrest 21712 . . . 4 (((𝑘Gen‘𝐽) ∈ Top ∧ 𝐽 ⊆ (𝑘Gen‘𝐽)) → (𝐽t 𝐾) ⊆ ((𝑘Gen‘𝐽) ↾t 𝐾))
2918, 27, 28syl2an2r 681 . . 3 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → (𝐽t 𝐾) ⊆ ((𝑘Gen‘𝐽) ↾t 𝐾))
30 eqid 2818 . . . 4 ((𝑘Gen‘𝐽) ↾t 𝐾) = ((𝑘Gen‘𝐽) ↾t 𝐾)
3130sscmp 21941 . . 3 (((𝐽t 𝐾) ∈ (TopOn‘ ((𝑘Gen‘𝐽) ↾t 𝐾)) ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp ∧ (𝐽t 𝐾) ⊆ ((𝑘Gen‘𝐽) ↾t 𝐾)) → (𝐽t 𝐾) ∈ Comp)
3224, 25, 29, 31syl3anc 1363 . 2 ((𝐽 ∈ Top ∧ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Comp)
333, 32impbida 797 1 (𝐽 ∈ Top → ((𝐽t 𝐾) ∈ Comp ↔ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  wss 3933   cuni 4830  cfv 6348  (class class class)co 7145  t crest 16682  Topctop 21429  TopOnctopon 21446  Compccmp 21922  𝑘Genckgen 22069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-oadd 8095  df-er 8278  df-en 8498  df-fin 8501  df-fi 8863  df-rest 16684  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cmp 21923  df-kgen 22070
This theorem is referenced by:  kgenidm  22083
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