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Theorem kqcld 23743
Description: The topological indistinguishability map is a closed map. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqcld ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹𝑈) ∈ (Clsd‘(KQ‘𝐽)))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem kqcld
StepHypRef Expression
1 imassrn 6089 . . . 4 (𝐹𝑈) ⊆ ran 𝐹
21a1i 11 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹𝑈) ⊆ ran 𝐹)
3 kqval.2 . . . . 5 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
43kqcldsat 23741 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) = 𝑈)
5 simpr 484 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ (Clsd‘𝐽))
64, 5eqeltrd 2841 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) ∈ (Clsd‘𝐽))
73kqffn 23733 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
8 dffn4 6826 . . . . . 6 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
97, 8sylib 218 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋onto→ran 𝐹)
10 qtopcld 23721 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → ((𝐹𝑈) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝐹𝑈) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑈)) ∈ (Clsd‘𝐽))))
119, 10mpdan 687 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → ((𝐹𝑈) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝐹𝑈) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑈)) ∈ (Clsd‘𝐽))))
1211adantr 480 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ((𝐹𝑈) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝐹𝑈) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑈)) ∈ (Clsd‘𝐽))))
132, 6, 12mpbir2and 713 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹𝑈) ∈ (Clsd‘(𝐽 qTop 𝐹)))
143kqval 23734 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
1514adantr 480 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
1615fveq2d 6910 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (Clsd‘(KQ‘𝐽)) = (Clsd‘(𝐽 qTop 𝐹)))
1713, 16eleqtrrd 2844 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹𝑈) ∈ (Clsd‘(KQ‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {crab 3436  wss 3951  cmpt 5225  ccnv 5684  ran crn 5686  cima 5688   Fn wfn 6556  ontowfo 6559  cfv 6561  (class class class)co 7431   qTop cqtop 17548  TopOnctopon 22916  Clsdccld 23024  KQckq 23701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17552  df-top 22900  df-topon 22917  df-cld 23027  df-kq 23702
This theorem is referenced by:  kqreglem1  23749  kqnrmlem1  23751  kqnrmlem2  23752
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