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Theorem kqt0 22070
Description: The Kolmogorov quotient is T0 even if the original topology is not. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
kqt0 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2)

Proof of Theorem kqt0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 toptopon2 21242 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
2 eqid 2772 . . . 4 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
32kqt0lem 22060 . . 3 (𝐽 ∈ (TopOn‘ 𝐽) → (KQ‘𝐽) ∈ Kol2)
41, 3sylbi 209 . 2 (𝐽 ∈ Top → (KQ‘𝐽) ∈ Kol2)
5 t0top 21653 . . 3 ((KQ‘𝐽) ∈ Kol2 → (KQ‘𝐽) ∈ Top)
6 kqtop 22069 . . 3 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
75, 6sylibr 226 . 2 ((KQ‘𝐽) ∈ Kol2 → 𝐽 ∈ Top)
84, 7impbii 201 1 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wcel 2050  {crab 3086   cuni 4708  cmpt 5004  cfv 6185  Topctop 21217  TopOnctopon 21234  Kol2ct0 21630  KQckq 22017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-qtop 16634  df-top 21218  df-topon 21235  df-t0 21637  df-kq 22018
This theorem is referenced by:  kqf  22071  kqhmph  22143
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