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Theorem kqf 22349
 Description: The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
kqf KQ:Top⟶Kol2

Proof of Theorem kqf
Dummy variables 𝑥 𝑦 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 7183 . . 3 (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})) ∈ V
2 df-kq 22296 . . 3 KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})))
31, 2fnmpti 6485 . 2 KQ Fn Top
4 kqt0 22348 . . . 4 (𝑥 ∈ Top ↔ (KQ‘𝑥) ∈ Kol2)
54biimpi 218 . . 3 (𝑥 ∈ Top → (KQ‘𝑥) ∈ Kol2)
65rgen 3148 . 2 𝑥 ∈ Top (KQ‘𝑥) ∈ Kol2
7 ffnfv 6876 . 2 (KQ:Top⟶Kol2 ↔ (KQ Fn Top ∧ ∀𝑥 ∈ Top (KQ‘𝑥) ∈ Kol2))
83, 6, 7mpbir2an 709 1 KQ:Top⟶Kol2
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2110  ∀wral 3138  {crab 3142  ∪ cuni 4831   ↦ cmpt 5138   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   qTop cqtop 16770  Topctop 21495  Kol2ct0 21908  KQckq 22295 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-qtop 16774  df-top 21496  df-topon 21513  df-t0 21915  df-kq 22296 This theorem is referenced by: (None)
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