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Theorem kqopn 23625
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 6068 . . . 4 (𝐹𝑈) ⊆ ran 𝐹
21a1i 11 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ⊆ ran 𝐹)
3 kqval.2 . . . . 5 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
43kqsat 23622 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) = 𝑈)
5 simpr 484 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝑈𝐽)
64, 5eqeltrd 2828 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) ∈ 𝐽)
73kqffn 23616 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
8 dffn4 6811 . . . . . 6 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
97, 8sylib 217 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋onto→ran 𝐹)
109adantr 480 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → 𝐹:𝑋onto→ran 𝐹)
11 elqtop3 23594 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → ((𝐹𝑈) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑈) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑈)) ∈ 𝐽)))
1210, 11syldan 590 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑈) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑈)) ∈ 𝐽)))
132, 6, 12mpbir2and 712 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ∈ (𝐽 qTop 𝐹))
143kqval 23617 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
1514adantr 480 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
1613, 15eleqtrrd 2831 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ∈ (KQ‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  {crab 3427  wss 3944  cmpt 5225  ccnv 5671  ran crn 5673  cima 5675   Fn wfn 6537  ontowfo 6540  cfv 6542  (class class class)co 7414   qTop cqtop 17476  TopOnctopon 22799  KQckq 23584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-qtop 17480  df-topon 22800  df-kq 23585
This theorem is referenced by:  kqt0lem  23627  isr0  23628  regr1lem  23630  kqreglem1  23632  kqreglem2  23633  kqnrmlem1  23634  kqnrmlem2  23635
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