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Theorem topnval 16289
 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 6344 . . . . . 6 (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
2 topnval.2 . . . . . 6 𝐽 = (TopSet‘𝑊)
31, 2syl6eqr 2804 . . . . 5 (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
4 fveq2 6344 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5 topnval.1 . . . . . 6 𝐵 = (Base‘𝑊)
64, 5syl6eqr 2804 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
73, 6oveq12d 6823 . . . 4 (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽t 𝐵))
8 df-topn 16278 . . . 4 TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
9 ovex 6833 . . . 4 (𝐽t 𝐵) ∈ V
107, 8, 9fvmpt 6436 . . 3 (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽t 𝐵))
1110eqcomd 2758 . 2 (𝑊 ∈ V → (𝐽t 𝐵) = (TopOpen‘𝑊))
12 0rest 16284 . . 3 (∅ ↾t 𝐵) = ∅
13 fvprc 6338 . . . . 5 𝑊 ∈ V → (TopSet‘𝑊) = ∅)
142, 13syl5eq 2798 . . . 4 𝑊 ∈ V → 𝐽 = ∅)
1514oveq1d 6820 . . 3 𝑊 ∈ V → (𝐽t 𝐵) = (∅ ↾t 𝐵))
16 fvprc 6338 . . 3 𝑊 ∈ V → (TopOpen‘𝑊) = ∅)
1712, 15, 163eqtr4a 2812 . 2 𝑊 ∈ V → (𝐽t 𝐵) = (TopOpen‘𝑊))
1811, 17pm2.61i 176 1 (𝐽t 𝐵) = (TopOpen‘𝑊)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1624   ∈ wcel 2131  Vcvv 3332  ∅c0 4050  ‘cfv 6041  (class class class)co 6805  Basecbs 16051  TopSetcts 16141   ↾t crest 16275  TopOpenctopn 16276 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-rest 16277  df-topn 16278 This theorem is referenced by:  topnid  16290  topnpropd  16291  oppgtopn  17975  symgtopn  18017  mgptopn  18690  resstopn  21184  prdstopn  21625  tuslem  22264  xrge0tsms  22830  om1opn  23028  xrge0tsmsd  30086  xrge0tmdOLD  30292
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