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Theorem topnpropd 16706
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 7157 . 2 (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿)))
4 eqid 2801 . . 3 (Base‘𝐾) = (Base‘𝐾)
5 eqid 2801 . . 3 (TopSet‘𝐾) = (TopSet‘𝐾)
64, 5topnval 16704 . 2 ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
7 eqid 2801 . . 3 (Base‘𝐿) = (Base‘𝐿)
8 eqid 2801 . . 3 (TopSet‘𝐿) = (TopSet‘𝐿)
97, 8topnval 16704 . 2 ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿)
103, 6, 93eqtr3g 2859 1 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  cfv 6328  (class class class)co 7139  Basecbs 16479  TopSetcts 16567  t crest 16690  TopOpenctopn 16691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-rest 16692  df-topn 16693
This theorem is referenced by:  sratopn  19954  tpsprop2d  21548  nrgtrg  23300  zhmnrg  31322
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