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Theorem topnpropgd 12193
 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 5801 . 2 (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿)))
4 topnpropgd.k . . 3 (𝜑𝐾𝑉)
5 eqid 2140 . . . 4 (Base‘𝐾) = (Base‘𝐾)
6 eqid 2140 . . . 4 (TopSet‘𝐾) = (TopSet‘𝐾)
75, 6topnvalg 12191 . . 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 2140 . . . 4 (Base‘𝐿) = (Base‘𝐿)
11 eqid 2140 . . . 4 (TopSet‘𝐿) = (TopSet‘𝐿)
1210, 11topnvalg 12191 . . 3 (𝐿𝑊 → ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿))
139, 12syl 14 . 2 (𝜑 → ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿))
143, 8, 133eqtr3d 2181 1 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  ‘cfv 5132  (class class class)co 5783  Basecbs 12018  TopSetcts 12086   ↾t crest 12179  TopOpenctopn 12180 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-cnex 7755  ax-resscn 7756  ax-1re 7758  ax-addrcl 7761 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-ov 5786  df-oprab 5787  df-mpo 5788  df-1st 6047  df-2nd 6048  df-inn 8765  df-2 8823  df-3 8824  df-4 8825  df-5 8826  df-6 8827  df-7 8828  df-8 8829  df-9 8830  df-ndx 12021  df-slot 12022  df-base 12024  df-tset 12099  df-rest 12181  df-topn 12182 This theorem is referenced by: (None)
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