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Theorem t0kq 21602
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 473 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐽 ∈ (TopOn‘𝑋))
2 t0kq.1 . . . . . 6 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
32ist0-4 21513 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋1-1→V))
43biimpa 501 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹:𝑋1-1→V)
51, 4qtopf1 21600 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))
62kqval 21510 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
76adantr 481 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
87oveq2d 6651 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → (𝐽Homeo(KQ‘𝐽)) = (𝐽Homeo(𝐽 qTop 𝐹)))
95, 8eleqtrrd 2702 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹 ∈ (𝐽Homeo(KQ‘𝐽)))
10 hmphi 21561 . . . . 5 (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → 𝐽 ≃ (KQ‘𝐽))
11 hmphsym 21566 . . . . 5 (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
1210, 11syl 17 . . . 4 (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → (KQ‘𝐽) ≃ 𝐽)
132kqt0lem 21520 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
14 t0hmph 21574 . . . 4 ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ Kol2 → 𝐽 ∈ Kol2))
1512, 13, 14syl2im 40 . . 3 (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Kol2))
1615impcom 446 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))) → 𝐽 ∈ Kol2)
179, 16impbida 876 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  {crab 2913  Vcvv 3195   class class class wbr 4644  cmpt 4720  1-1wf1 5873  cfv 5876  (class class class)co 6635   qTop cqtop 16144  TopOnctopon 20696  Kol2ct0 21091  KQckq 21477  Homeochmeo 21537  chmph 21538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-1o 7545  df-map 7844  df-qtop 16148  df-top 20680  df-topon 20697  df-cn 21012  df-t0 21098  df-kq 21478  df-hmeo 21539  df-hmph 21540
This theorem is referenced by:  kqhmph  21603
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