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

Theorem kqid 21753
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 21750 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3 qtopid 21730 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
42, 3mpdan 705 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
51kqval 21751 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
65oveq2d 6830 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 Cn (KQ‘𝐽)) = (𝐽 Cn (𝐽 qTop 𝐹)))
74, 6eleqtrrd 2842 1 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  {crab 3054  cmpt 4881   Fn wfn 6044  cfv 6049  (class class class)co 6814   qTop cqtop 16385  TopOnctopon 20937   Cn ccn 21250  KQckq 21718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-qtop 16389  df-top 20921  df-topon 20938  df-cn 21253  df-kq 21719
This theorem is referenced by:  isr0  21762  r0cld  21763  kqreglem1  21766  kqreglem2  21767  kqnrmlem1  21768  kqnrmlem2  21769
  Copyright terms: Public domain W3C validator