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Theorem topnval 17379
Description: Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
topnval.1 𝐵 = (Base‘𝑊)
topnval.2 𝐽 = (TopSet‘𝑊)
Assertion
Ref Expression
topnval (𝐽t 𝐵) = (TopOpen‘𝑊)

Proof of Theorem topnval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6881 . . . . . 6 (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
2 topnval.2 . . . . . 6 𝐽 = (TopSet‘𝑊)
31, 2eqtr4di 2782 . . . . 5 (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
4 fveq2 6881 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5 topnval.1 . . . . . 6 𝐵 = (Base‘𝑊)
64, 5eqtr4di 2782 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
73, 6oveq12d 7419 . . . 4 (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽t 𝐵))
8 df-topn 17368 . . . 4 TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
9 ovex 7434 . . . 4 (𝐽t 𝐵) ∈ V
107, 8, 9fvmpt 6988 . . 3 (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽t 𝐵))
1110eqcomd 2730 . 2 (𝑊 ∈ V → (𝐽t 𝐵) = (TopOpen‘𝑊))
12 0rest 17374 . . 3 (∅ ↾t 𝐵) = ∅
13 fvprc 6873 . . . . 5 𝑊 ∈ V → (TopSet‘𝑊) = ∅)
142, 13eqtrid 2776 . . . 4 𝑊 ∈ V → 𝐽 = ∅)
1514oveq1d 7416 . . 3 𝑊 ∈ V → (𝐽t 𝐵) = (∅ ↾t 𝐵))
16 fvprc 6873 . . 3 𝑊 ∈ V → (TopOpen‘𝑊) = ∅)
1712, 15, 163eqtr4a 2790 . 2 𝑊 ∈ V → (𝐽t 𝐵) = (TopOpen‘𝑊))
1811, 17pm2.61i 182 1 (𝐽t 𝐵) = (TopOpen‘𝑊)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  Vcvv 3466  c0 4314  cfv 6533  (class class class)co 7401  Basecbs 17143  TopSetcts 17202  t crest 17365  TopOpenctopn 17366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-rest 17367  df-topn 17368
This theorem is referenced by:  topnid  17380  topnpropd  17381  efmndtopn  18798  oppgtopn  19262  symgtopn  19316  mgptopn  20041  resstopn  23012  prdstopn  23454  tuslem  24093  tuslemOLD  24094  xrge0tsms  24672  om1opn  24885  xrge0tsmsd  32677  xrge0tmdALT  33415
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