Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqid Structured version   Visualization version   GIF version

Theorem kqid 22330
 Description: The topological indistinguishability map is a continuous function into the Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqid (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqid
StepHypRef Expression
1 kqval.2 . . . 4 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqffn 22327 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3 qtopid 22307 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
42, 3mpdan 685 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
51kqval 22328 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
65oveq2d 7166 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 Cn (KQ‘𝐽)) = (𝐽 Cn (𝐽 qTop 𝐹)))
74, 6eleqtrrd 2916 1 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  {crab 3142   ↦ cmpt 5138   Fn wfn 6344  ‘cfv 6349  (class class class)co 7150   qTop cqtop 16770  TopOnctopon 21512   Cn ccn 21826  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-map 8402  df-qtop 16774  df-top 21496  df-topon 21513  df-cn 21829  df-kq 22296 This theorem is referenced by:  isr0  22339  r0cld  22340  kqreglem1  22343  kqreglem2  22344  kqnrmlem1  22345  kqnrmlem2  22346
 Copyright terms: Public domain W3C validator