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Theorem kqtop 21596
Description: The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
kqtop (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)

Proof of Theorem kqtop
Dummy variables 𝑥 𝑦 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . . . 5 𝐽 = 𝐽
21toptopon 20770 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
3 eqid 2651 . . . . 5 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
43kqtopon 21578 . . . 4 (𝐽 ∈ (TopOn‘ 𝐽) → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
52, 4sylbi 207 . . 3 (𝐽 ∈ Top → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
6 topontop 20766 . . 3 ((KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})) → (KQ‘𝐽) ∈ Top)
75, 6syl 17 . 2 (𝐽 ∈ Top → (KQ‘𝐽) ∈ Top)
8 0opn 20757 . . . 4 ((KQ‘𝐽) ∈ Top → ∅ ∈ (KQ‘𝐽))
9 elfvdm 6258 . . . 4 (∅ ∈ (KQ‘𝐽) → 𝐽 ∈ dom KQ)
108, 9syl 17 . . 3 ((KQ‘𝐽) ∈ Top → 𝐽 ∈ dom KQ)
11 ovex 6718 . . . 4 (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})) ∈ V
12 df-kq 21545 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})))
1311, 12dmmpti 6061 . . 3 dom KQ = Top
1410, 13syl6eleq 2740 . 2 ((KQ‘𝐽) ∈ Top → 𝐽 ∈ Top)
157, 14impbii 199 1 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2030  {crab 2945  c0 3948   cuni 4468  cmpt 4762  dom cdm 5143  ran crn 5144  cfv 5926  (class class class)co 6690   qTop cqtop 16210  Topctop 20746  TopOnctopon 20763  KQckq 21544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-qtop 16214  df-top 20747  df-topon 20764  df-kq 21545
This theorem is referenced by:  kqt0  21597  kqreg  21602  kqnrm  21603
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