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Theorem t0kq 22412
Description: A topological space is T0 iff the quotient map is a homeomorphism onto the space's Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
t0kq.1 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
t0kq (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem t0kq
StepHypRef Expression
1 simpl 486 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐽 ∈ (TopOn‘𝑋))
2 t0kq.1 . . . . . 6 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
32ist0-4 22323 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋1-1→V))
43biimpa 480 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹:𝑋1-1→V)
51, 4qtopf1 22410 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))
62kqval 22320 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
76adantr 484 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
87oveq2d 7154 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → (𝐽Homeo(KQ‘𝐽)) = (𝐽Homeo(𝐽 qTop 𝐹)))
95, 8eleqtrrd 2919 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹 ∈ (𝐽Homeo(KQ‘𝐽)))
10 hmphi 22371 . . . . 5 (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → 𝐽 ≃ (KQ‘𝐽))
11 hmphsym 22376 . . . . 5 (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
1210, 11syl 17 . . . 4 (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → (KQ‘𝐽) ≃ 𝐽)
132kqt0lem 22330 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
14 t0hmph 22384 . . . 4 ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ Kol2 → 𝐽 ∈ Kol2))
1512, 13, 14syl2im 40 . . 3 (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Kol2))
1615impcom 411 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))) → 𝐽 ∈ Kol2)
179, 16impbida 800 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {crab 3136  Vcvv 3479   class class class wbr 5047  cmpt 5127  1-1wf1 6333  cfv 6336  (class class class)co 7138   qTop cqtop 16765  TopOnctopon 21504  Kol2ct0 21900  KQckq 22287  Homeochmeo 22347  chmph 22348
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 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673  df-1o 8085  df-map 8391  df-qtop 16769  df-top 21488  df-topon 21505  df-cn 21821  df-t0 21907  df-kq 22288  df-hmeo 22349  df-hmph 22350
This theorem is referenced by:  kqhmph  22413
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