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Theorem topnval 17479
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 6906 . . . . . 6 (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
2 topnval.2 . . . . . 6 𝐽 = (TopSet‘𝑊)
31, 2eqtr4di 2795 . . . . 5 (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
4 fveq2 6906 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5 topnval.1 . . . . . 6 𝐵 = (Base‘𝑊)
64, 5eqtr4di 2795 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
73, 6oveq12d 7449 . . . 4 (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽t 𝐵))
8 df-topn 17468 . . . 4 TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
9 ovex 7464 . . . 4 (𝐽t 𝐵) ∈ V
107, 8, 9fvmpt 7016 . . 3 (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽t 𝐵))
1110eqcomd 2743 . 2 (𝑊 ∈ V → (𝐽t 𝐵) = (TopOpen‘𝑊))
12 0rest 17474 . . 3 (∅ ↾t 𝐵) = ∅
13 fvprc 6898 . . . . 5 𝑊 ∈ V → (TopSet‘𝑊) = ∅)
142, 13eqtrid 2789 . . . 4 𝑊 ∈ V → 𝐽 = ∅)
1514oveq1d 7446 . . 3 𝑊 ∈ V → (𝐽t 𝐵) = (∅ ↾t 𝐵))
16 fvprc 6898 . . 3 𝑊 ∈ V → (TopOpen‘𝑊) = ∅)
1712, 15, 163eqtr4a 2803 . 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 1540  wcel 2108  Vcvv 3480  c0 4333  cfv 6561  (class class class)co 7431  Basecbs 17247  TopSetcts 17303  t crest 17465  TopOpenctopn 17466
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-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-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-1st 8014  df-2nd 8015  df-rest 17467  df-topn 17468
This theorem is referenced by:  topnid  17480  topnpropd  17481  efmndtopn  18896  oppgtopn  19372  symgtopn  19424  mgptopn  20145  resstopn  23194  prdstopn  23636  tuslem  24275  tuslemOLD  24276  xrge0tsms  24856  om1opn  25069  xrge0tsmsd  33065  xrge0tmdALT  33945
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