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Theorem kqtop 23240
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 22411 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
2 eqid 2732 . . . . 5 (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦}) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
32kqtopon 23222 . . . 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 22406 . . 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 22397 . . . 4 ((KQβ€˜π½) ∈ Top β†’ βˆ… ∈ (KQβ€˜π½))
8 elfvdm 6925 . . . 4 (βˆ… ∈ (KQβ€˜π½) β†’ 𝐽 ∈ dom KQ)
97, 8syl 17 . . 3 ((KQβ€˜π½) ∈ Top β†’ 𝐽 ∈ dom KQ)
10 ovex 7438 . . . 4 (𝑗 qTop (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ π‘₯ ∈ 𝑦})) ∈ V
11 df-kq 23189 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ π‘₯ ∈ 𝑦})))
1210, 11dmmpti 6691 . . 3 dom KQ = Top
139, 12eleqtrdi 2843 . 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 2106  {crab 3432  βˆ…c0 4321  βˆͺ cuni 4907   ↦ cmpt 5230  dom cdm 5675  ran crn 5676  β€˜cfv 6540  (class class class)co 7405   qTop cqtop 17445  Topctop 22386  TopOnctopon 22403  KQckq 23188
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-qtop 17449  df-top 22387  df-topon 22404  df-kq 23189
This theorem is referenced by:  kqt0  23241  kqreg  23246  kqnrm  23247
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