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Theorem kqopn 21347
 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 5396 . . . 4 (𝐹𝑈) ⊆ ran 𝐹
21a1i 11 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ⊆ ran 𝐹)
3 kqval.2 . . . . 5 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
43kqsat 21344 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) = 𝑈)
5 simpr 476 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈𝐽)
64, 5eqeltrd 2688 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) ∈ 𝐽)
73kqffn 21338 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
8 dffn4 6034 . . . . . 6 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
97, 8sylib 207 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋onto→ran 𝐹)
109adantr 480 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝐹:𝑋onto→ran 𝐹)
11 elqtop3 21316 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → ((𝐹𝑈) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑈) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑈)) ∈ 𝐽)))
1210, 11syldan 486 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑈) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑈)) ∈ 𝐽)))
132, 6, 12mpbir2and 959 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ∈ (𝐽 qTop 𝐹))
143kqval 21339 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
1514adantr 480 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
1613, 15eleqtrrd 2691 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ∈ (KQ‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ⊆ wss 3540   ↦ cmpt 4643  ◡ccnv 5037  ran crn 5039   “ cima 5041   Fn wfn 5799  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   qTop cqtop 15986  TopOnctopon 20518  KQckq 21306 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-qtop 15990  df-topon 20523  df-kq 21307 This theorem is referenced by:  kqt0lem  21349  isr0  21350  regr1lem  21352  kqreglem1  21354  kqreglem2  21355  kqnrmlem1  21356  kqnrmlem2  21357
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