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Theorem topnpropgd 12061
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 5760 . 2 (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿)))
4 topnpropgd.k . . 3 (𝜑𝐾𝑉)
5 eqid 2117 . . . 4 (Base‘𝐾) = (Base‘𝐾)
6 eqid 2117 . . . 4 (TopSet‘𝐾) = (TopSet‘𝐾)
75, 6topnvalg 12059 . . 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 2117 . . . 4 (Base‘𝐿) = (Base‘𝐿)
11 eqid 2117 . . . 4 (TopSet‘𝐿) = (TopSet‘𝐿)
1210, 11topnvalg 12059 . . 3 (𝐿𝑊 → ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿))
139, 12syl 14 . 2 (𝜑 → ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿))
143, 8, 133eqtr3d 2158 1 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465  cfv 5093  (class class class)co 5742  Basecbs 11886  TopSetcts 11954  t crest 12047  TopOpenctopn 12048
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-5 8750  df-6 8751  df-7 8752  df-8 8753  df-9 8754  df-ndx 11889  df-slot 11890  df-base 11892  df-tset 11967  df-rest 12049  df-topn 12050
This theorem is referenced by: (None)
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