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Theorem kqtopon 21908
Description: The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqtopon (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqtopon
StepHypRef Expression
1 kqval.2 . . 3 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqval 21907 . 2 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
31kqffn 21906 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
4 dffn4 6363 . . . 4 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
53, 4sylib 210 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋onto→ran 𝐹)
6 qtoptopon 21885 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
75, 6mpdan 678 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
82, 7eqeltrd 2906 1 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  {crab 3121  cmpt 4954  ran crn 5347   Fn wfn 6122  ontowfo 6125  cfv 6127  (class class class)co 6910   qTop cqtop 16523  TopOnctopon 21092  KQckq 21874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-qtop 16527  df-top 21076  df-topon 21093  df-kq 21875
This theorem is referenced by:  kqt0lem  21917  isr0  21918  r0cld  21919  regr1lem2  21921  kqreglem1  21922  kqreglem2  21923  kqnrmlem1  21924  kqnrmlem2  21925  kqtop  21926
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