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Theorem topnpropgd 12719
Description: The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.)
Hypotheses
Ref Expression
topnpropd.1 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
topnpropd.2 (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))
topnpropgd.k (𝜑𝐾𝑉)
topnpropgd.l (𝜑𝐿𝑊)
Assertion
Ref Expression
topnpropgd (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Proof of Theorem topnpropgd
StepHypRef Expression
1 topnpropd.2 . . 3 (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))
2 topnpropd.1 . . 3 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
31, 2oveq12d 5906 . 2 (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿)))
4 topnpropgd.k . . 3 (𝜑𝐾𝑉)
5 eqid 2187 . . . 4 (Base‘𝐾) = (Base‘𝐾)
6 eqid 2187 . . . 4 (TopSet‘𝐾) = (TopSet‘𝐾)
75, 6topnvalg 12717 . . 3 (𝐾𝑉 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾))
84, 7syl 14 . 2 (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾))
9 topnpropgd.l . . 3 (𝜑𝐿𝑊)
10 eqid 2187 . . . 4 (Base‘𝐿) = (Base‘𝐿)
11 eqid 2187 . . . 4 (TopSet‘𝐿) = (TopSet‘𝐿)
1210, 11topnvalg 12717 . . 3 (𝐿𝑊 → ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿))
139, 12syl 14 . 2 (𝜑 → ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿))
143, 8, 133eqtr3d 2228 1 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wcel 2158  cfv 5228  (class class class)co 5888  Basecbs 12475  TopSetcts 12556  t crest 12705  TopOpenctopn 12706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-5 8994  df-6 8995  df-7 8996  df-8 8997  df-9 8998  df-ndx 12478  df-slot 12479  df-base 12481  df-tset 12569  df-rest 12707  df-topn 12708
This theorem is referenced by:  sratopng  13631
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