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Theorem topnpropd 15920
 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 6567 . 2 (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿)))
4 eqid 2610 . . 3 (Base‘𝐾) = (Base‘𝐾)
5 eqid 2610 . . 3 (TopSet‘𝐾) = (TopSet‘𝐾)
64, 5topnval 15918 . 2 ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
7 eqid 2610 . . 3 (Base‘𝐿) = (Base‘𝐿)
8 eqid 2610 . . 3 (TopSet‘𝐿) = (TopSet‘𝐿)
97, 8topnval 15918 . 2 ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿)
103, 6, 93eqtr3g 2667 1 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  TopSetcts 15774   ↾t crest 15904  TopOpenctopn 15905 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-rest 15906  df-topn 15907 This theorem is referenced by:  sratopn  19006  tpsprop2d  20556  nrgtrg  22304  zhmnrg  29339
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