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Theorem kqtop 22359
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 21532 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
2 eqid 2824 . . . . 5 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
32kqtopon 22341 . . . 4 (𝐽 ∈ (TopOn‘ 𝐽) → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
41, 3sylbi 220 . . 3 (𝐽 ∈ Top → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
5 topontop 21527 . . 3 ((KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})) → (KQ‘𝐽) ∈ Top)
64, 5syl 17 . 2 (𝐽 ∈ Top → (KQ‘𝐽) ∈ Top)
7 0opn 21518 . . . 4 ((KQ‘𝐽) ∈ Top → ∅ ∈ (KQ‘𝐽))
8 elfvdm 6695 . . . 4 (∅ ∈ (KQ‘𝐽) → 𝐽 ∈ dom KQ)
97, 8syl 17 . . 3 ((KQ‘𝐽) ∈ Top → 𝐽 ∈ dom KQ)
10 ovex 7184 . . . 4 (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})) ∈ V
11 df-kq 22308 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})))
1210, 11dmmpti 6483 . . 3 dom KQ = Top
139, 12eleqtrdi 2926 . 2 ((KQ‘𝐽) ∈ Top → 𝐽 ∈ Top)
146, 13impbii 212 1 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2115  {crab 3137  c0 4276   cuni 4824  cmpt 5133  dom cdm 5543  ran crn 5544  cfv 6345  (class class class)co 7151   qTop cqtop 16778  Topctop 21507  TopOnctopon 21524  KQckq 22307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-qtop 16782  df-top 21508  df-topon 21525  df-kq 22308
This theorem is referenced by:  kqt0  22360  kqreg  22365  kqnrm  22366
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