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Theorem kqtop 23662
Description: The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
kqtop (𝐽 ∈ Top ↔ (KQβ€˜π½) ∈ Top)

Proof of Theorem kqtop
Dummy variables π‘₯ 𝑦 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 toptopon2 22833 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
2 eqid 2728 . . . . 5 (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦}) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
32kqtopon 23644 . . . 4 (𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) β†’ (KQβ€˜π½) ∈ (TopOnβ€˜ran (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})))
41, 3sylbi 216 . . 3 (𝐽 ∈ Top β†’ (KQβ€˜π½) ∈ (TopOnβ€˜ran (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})))
5 topontop 22828 . . 3 ((KQβ€˜π½) ∈ (TopOnβ€˜ran (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})) β†’ (KQβ€˜π½) ∈ Top)
64, 5syl 17 . 2 (𝐽 ∈ Top β†’ (KQβ€˜π½) ∈ Top)
7 0opn 22819 . . . 4 ((KQβ€˜π½) ∈ Top β†’ βˆ… ∈ (KQβ€˜π½))
8 elfvdm 6934 . . . 4 (βˆ… ∈ (KQβ€˜π½) β†’ 𝐽 ∈ dom KQ)
97, 8syl 17 . . 3 ((KQβ€˜π½) ∈ Top β†’ 𝐽 ∈ dom KQ)
10 ovex 7453 . . . 4 (𝑗 qTop (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ π‘₯ ∈ 𝑦})) ∈ V
11 df-kq 23611 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ π‘₯ ∈ 𝑦})))
1210, 11dmmpti 6699 . . 3 dom KQ = Top
139, 12eleqtrdi 2839 . 2 ((KQβ€˜π½) ∈ Top β†’ 𝐽 ∈ Top)
146, 13impbii 208 1 (𝐽 ∈ Top ↔ (KQβ€˜π½) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∈ wcel 2099  {crab 3429  βˆ…c0 4323  βˆͺ cuni 4908   ↦ cmpt 5231  dom cdm 5678  ran crn 5679  β€˜cfv 6548  (class class class)co 7420   qTop cqtop 17485  Topctop 22808  TopOnctopon 22825  KQckq 23610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-qtop 17489  df-top 22809  df-topon 22826  df-kq 23611
This theorem is referenced by:  kqt0  23663  kqreg  23668  kqnrm  23669
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