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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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