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Theorem kqtop 21877
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 2799 . . . . 5 𝐽 = 𝐽
21toptopon 21050 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
3 eqid 2799 . . . . 5 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
43kqtopon 21859 . . . 4 (𝐽 ∈ (TopOn‘ 𝐽) → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
52, 4sylbi 209 . . 3 (𝐽 ∈ Top → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})))
6 topontop 21046 . . 3 ((KQ‘𝐽) ∈ (TopOn‘ran (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})) → (KQ‘𝐽) ∈ Top)
75, 6syl 17 . 2 (𝐽 ∈ Top → (KQ‘𝐽) ∈ Top)
8 0opn 21037 . . . 4 ((KQ‘𝐽) ∈ Top → ∅ ∈ (KQ‘𝐽))
9 elfvdm 6443 . . . 4 (∅ ∈ (KQ‘𝐽) → 𝐽 ∈ dom KQ)
108, 9syl 17 . . 3 ((KQ‘𝐽) ∈ Top → 𝐽 ∈ dom KQ)
11 ovex 6910 . . . 4 (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})) ∈ V
12 df-kq 21826 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})))
1311, 12dmmpti 6234 . . 3 dom KQ = Top
1410, 13syl6eleq 2888 . 2 ((KQ‘𝐽) ∈ Top → 𝐽 ∈ Top)
157, 14impbii 201 1 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wcel 2157  {crab 3093  c0 4115   cuni 4628  cmpt 4922  dom cdm 5312  ran crn 5313  cfv 6101  (class class class)co 6878   qTop cqtop 16478  Topctop 21026  TopOnctopon 21043  KQckq 21825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-qtop 16482  df-top 21027  df-topon 21044  df-kq 21826
This theorem is referenced by:  kqt0  21878  kqreg  21883  kqnrm  21884
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