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Theorem topnpropd 17481
Description: The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.)
Hypotheses
Ref Expression
topnpropd.1 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
topnpropd.2 (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))
Assertion
Ref Expression
topnpropd (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Proof of Theorem topnpropd
StepHypRef Expression
1 topnpropd.2 . . 3 (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))
2 topnpropd.1 . . 3 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
31, 2oveq12d 7449 . 2 (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿)))
4 eqid 2737 . . 3 (Base‘𝐾) = (Base‘𝐾)
5 eqid 2737 . . 3 (TopSet‘𝐾) = (TopSet‘𝐾)
64, 5topnval 17479 . 2 ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
7 eqid 2737 . . 3 (Base‘𝐿) = (Base‘𝐿)
8 eqid 2737 . . 3 (TopSet‘𝐿) = (TopSet‘𝐿)
97, 8topnval 17479 . 2 ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿)
103, 6, 93eqtr3g 2800 1 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  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:  sratopn  21190  tpsprop2d  22945  nrgtrg  24711  zhmnrg  33966
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