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Theorem topnpropd 16299
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 6831 . 2 (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿)))
4 eqid 2760 . . 3 (Base‘𝐾) = (Base‘𝐾)
5 eqid 2760 . . 3 (TopSet‘𝐾) = (TopSet‘𝐾)
64, 5topnval 16297 . 2 ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
7 eqid 2760 . . 3 (Base‘𝐿) = (Base‘𝐿)
8 eqid 2760 . . 3 (TopSet‘𝐿) = (TopSet‘𝐿)
97, 8topnval 16297 . 2 ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿)
103, 6, 93eqtr3g 2817 1 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  cfv 6049  (class class class)co 6813  Basecbs 16059  TopSetcts 16149  t crest 16283  TopOpenctopn 16284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-rest 16285  df-topn 16286
This theorem is referenced by:  sratopn  19387  tpsprop2d  20945  nrgtrg  22695  zhmnrg  30320
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