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Theorem kqtop 21358
 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 2610 . . . . 5 𝐽 = 𝐽
21toptopon 20548 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
3 eqid 2610 . . . . 5 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
43kqtopon 21340 . . . 4 (𝐽 ∈ (TopOn‘ 𝐽) → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
52, 4sylbi 206 . . 3 (𝐽 ∈ Top → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
6 topontop 20541 . . 3 ((KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})) → (KQ‘𝐽) ∈ Top)
75, 6syl 17 . 2 (𝐽 ∈ Top → (KQ‘𝐽) ∈ Top)
8 0opn 20534 . . . 4 ((KQ‘𝐽) ∈ Top → ∅ ∈ (KQ‘𝐽))
9 elfvdm 6130 . . . 4 (∅ ∈ (KQ‘𝐽) → 𝐽 ∈ dom KQ)
108, 9syl 17 . . 3 ((KQ‘𝐽) ∈ Top → 𝐽 ∈ dom KQ)
11 ovex 6577 . . . 4 (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})) ∈ V
12 df-kq 21307 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})))
1311, 12dmmpti 5936 . . 3 dom KQ = Top
1410, 13syl6eleq 2698 . 2 ((KQ‘𝐽) ∈ Top → 𝐽 ∈ Top)
157, 14impbii 198 1 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977  {crab 2900  ∅c0 3874  ∪ cuni 4372   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  ‘cfv 5804  (class class class)co 6549   qTop cqtop 15986  Topctop 20517  TopOnctopon 20518  KQckq 21306 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-qtop 15990  df-top 20521  df-topon 20523  df-kq 21307 This theorem is referenced by:  kqt0  21359  kqreg  21364  kqnrm  21365
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