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Theorem kqopn 21739
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 5635 . . . 4 (𝐹𝑈) ⊆ ran 𝐹
21a1i 11 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ⊆ ran 𝐹)
3 kqval.2 . . . . 5 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
43kqsat 21736 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) = 𝑈)
5 simpr 479 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈𝐽)
64, 5eqeltrd 2839 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) ∈ 𝐽)
73kqffn 21730 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
8 dffn4 6282 . . . . . 6 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
97, 8sylib 208 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋onto→ran 𝐹)
109adantr 472 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝐹:𝑋onto→ran 𝐹)
11 elqtop3 21708 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → ((𝐹𝑈) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑈) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑈)) ∈ 𝐽)))
1210, 11syldan 488 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑈) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑈)) ∈ 𝐽)))
132, 6, 12mpbir2and 995 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ∈ (𝐽 qTop 𝐹))
143kqval 21731 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
1514adantr 472 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
1613, 15eleqtrrd 2842 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ∈ (KQ‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {crab 3054  wss 3715  cmpt 4881  ccnv 5265  ran crn 5267  cima 5269   Fn wfn 6044  ontowfo 6047  cfv 6049  (class class class)co 6813   qTop cqtop 16365  TopOnctopon 20917  KQckq 21698
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 7114
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 6816  df-oprab 6817  df-mpt2 6818  df-qtop 16369  df-topon 20918  df-kq 21699
This theorem is referenced by:  kqt0lem  21741  isr0  21742  regr1lem  21744  kqreglem1  21746  kqreglem2  21747  kqnrmlem1  21748  kqnrmlem2  21749
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