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Theorem kqhmph 22422
Description: A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
kqhmph (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))

Proof of Theorem kqhmph
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 t0top 21932 . . . . . 6 (𝐽 ∈ Kol2 → 𝐽 ∈ Top)
2 toptopon2 21521 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
31, 2sylib 221 . . . . 5 (𝐽 ∈ Kol2 → 𝐽 ∈ (TopOn‘ 𝐽))
4 eqid 2822 . . . . . 6 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
54t0kq 22421 . . . . 5 (𝐽 ∈ (TopOn‘ 𝐽) → (𝐽 ∈ Kol2 ↔ (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) ∈ (𝐽Homeo(KQ‘𝐽))))
63, 5syl 17 . . . 4 (𝐽 ∈ Kol2 → (𝐽 ∈ Kol2 ↔ (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) ∈ (𝐽Homeo(KQ‘𝐽))))
76ibi 270 . . 3 (𝐽 ∈ Kol2 → (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) ∈ (𝐽Homeo(KQ‘𝐽)))
8 hmphi 22380 . . 3 ((𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) ∈ (𝐽Homeo(KQ‘𝐽)) → 𝐽 ≃ (KQ‘𝐽))
97, 8syl 17 . 2 (𝐽 ∈ Kol2 → 𝐽 ≃ (KQ‘𝐽))
10 hmphsym 22385 . . 3 (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
11 hmphtop1 22382 . . . 4 (𝐽 ≃ (KQ‘𝐽) → 𝐽 ∈ Top)
12 kqt0 22349 . . . 4 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2)
1311, 12sylib 221 . . 3 (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ∈ Kol2)
14 t0hmph 22393 . . 3 ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ Kol2 → 𝐽 ∈ Kol2))
1510, 13, 14sylc 65 . 2 (𝐽 ≃ (KQ‘𝐽) → 𝐽 ∈ Kol2)
169, 15impbii 212 1 (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2114  {crab 3134   cuni 4813   class class class wbr 5042  cmpt 5122  cfv 6334  (class class class)co 7140  Topctop 21496  TopOnctopon 21513  Kol2ct0 21909  KQckq 22296  Homeochmeo 22356  chmph 22357
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-1o 8089  df-map 8395  df-qtop 16771  df-top 21497  df-topon 21514  df-cn 21830  df-t0 21916  df-kq 22297  df-hmeo 22358  df-hmph 22359
This theorem is referenced by:  ist1-5lem  22423  t1r0  22424
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