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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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