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Theorem kqtop 23768
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 toptopon2 22939 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
2 eqid 2734 . . . . 5 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
32kqtopon 23750 . . . 4 (𝐽 ∈ (TopOn‘ 𝐽) → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
41, 3sylbi 217 . . 3 (𝐽 ∈ Top → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
5 topontop 22934 . . 3 ((KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})) → (KQ‘𝐽) ∈ Top)
64, 5syl 17 . 2 (𝐽 ∈ Top → (KQ‘𝐽) ∈ Top)
7 0opn 22925 . . . 4 ((KQ‘𝐽) ∈ Top → ∅ ∈ (KQ‘𝐽))
8 elfvdm 6943 . . . 4 (∅ ∈ (KQ‘𝐽) → 𝐽 ∈ dom KQ)
97, 8syl 17 . . 3 ((KQ‘𝐽) ∈ Top → 𝐽 ∈ dom KQ)
10 ovex 7463 . . . 4 (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})) ∈ V
11 df-kq 23717 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})))
1210, 11dmmpti 6712 . . 3 dom KQ = Top
139, 12eleqtrdi 2848 . 2 ((KQ‘𝐽) ∈ Top → 𝐽 ∈ Top)
146, 13impbii 209 1 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2105  {crab 3432  c0 4338   cuni 4911  cmpt 5230  dom cdm 5688  ran crn 5689  cfv 6562  (class class class)co 7430   qTop cqtop 17549  Topctop 22914  TopOnctopon 22931  KQckq 23716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-qtop 17553  df-top 22915  df-topon 22932  df-kq 23717
This theorem is referenced by:  kqt0  23769  kqreg  23774  kqnrm  23775
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