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Theorem kqtop 23593
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 22764 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
2 eqid 2724 . . . . 5 (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦}) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
32kqtopon 23575 . . . 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 22759 . . 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 22750 . . . 4 ((KQβ€˜π½) ∈ Top β†’ βˆ… ∈ (KQβ€˜π½))
8 elfvdm 6919 . . . 4 (βˆ… ∈ (KQβ€˜π½) β†’ 𝐽 ∈ dom KQ)
97, 8syl 17 . . 3 ((KQβ€˜π½) ∈ Top β†’ 𝐽 ∈ dom KQ)
10 ovex 7435 . . . 4 (𝑗 qTop (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ π‘₯ ∈ 𝑦})) ∈ V
11 df-kq 23542 . . . 4 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ π‘₯ ∈ 𝑦})))
1210, 11dmmpti 6685 . . 3 dom KQ = Top
139, 12eleqtrdi 2835 . 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 2098  {crab 3424  βˆ…c0 4315  βˆͺ cuni 4900   ↦ cmpt 5222  dom cdm 5667  ran crn 5668  β€˜cfv 6534  (class class class)co 7402   qTop cqtop 17454  Topctop 22739  TopOnctopon 22756  KQckq 23541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-qtop 17458  df-top 22740  df-topon 22757  df-kq 23542
This theorem is referenced by:  kqt0  23594  kqreg  23599  kqnrm  23600
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