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Theorem kqopn 21957
Description: The topological indistinguishability map is an open map. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqopn ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ∈ (KQ‘𝐽))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem kqopn
StepHypRef Expression
1 imassrn 5733 . . . 4 (𝐹𝑈) ⊆ ran 𝐹
21a1i 11 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ⊆ ran 𝐹)
3 kqval.2 . . . . 5 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
43kqsat 21954 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) = 𝑈)
5 simpr 479 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈𝐽)
64, 5eqeltrd 2859 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) ∈ 𝐽)
73kqffn 21948 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
8 dffn4 6374 . . . . . 6 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
97, 8sylib 210 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋onto→ran 𝐹)
109adantr 474 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝐹:𝑋onto→ran 𝐹)
11 elqtop3 21926 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → ((𝐹𝑈) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑈) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑈)) ∈ 𝐽)))
1210, 11syldan 585 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑈) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑈)) ∈ 𝐽)))
132, 6, 12mpbir2and 703 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ∈ (𝐽 qTop 𝐹))
143kqval 21949 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
1514adantr 474 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
1613, 15eleqtrrd 2862 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ∈ (KQ‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  {crab 3094  wss 3792  cmpt 4967  ccnv 5356  ran crn 5358  cima 5360   Fn wfn 6132  ontowfo 6135  cfv 6137  (class class class)co 6924   qTop cqtop 16560  TopOnctopon 21133  KQckq 21916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-qtop 16564  df-topon 21134  df-kq 21917
This theorem is referenced by:  kqt0lem  21959  isr0  21960  regr1lem  21962  kqreglem1  21964  kqreglem2  21965  kqnrmlem1  21966  kqnrmlem2  21967
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