Theorem topnpropgd 12730

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	⊢ (𝜑 → 𝐾 ∈ 𝑉)
topnpropgd.l	⊢ (𝜑 → 𝐿 ∈ 𝑊)

Assertion

Ref	Expression
topnpropgd	⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Proof of Theorem topnpropgd

Step	Hyp	Ref	Expression
1		topnpropd.2	. . 3 ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))
2		topnpropd.1	. . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
3	1, 2	oveq12d 5909	. 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿)))
4		topnpropgd.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
5		eqid 2189	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		eqid 2189	. . . 4 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
7	5, 6	topnvalg 12728	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾))
8	4, 7	syl 14	. 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾))
9		topnpropgd.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ 𝑊)
10		eqid 2189	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
11		eqid 2189	. . . 4 ⊢ (TopSet‘𝐿) = (TopSet‘𝐿)
12	10, 11	topnvalg 12728	. . 3 ⊢ (𝐿 ∈ 𝑊 → ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿))
13	9, 12	syl 14	. 2 ⊢ (𝜑 → ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿))
14	3, 8, 13	3eqtr3d 2230	1 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2160  ‘cfv 5231  (class class class)co 5891  Basecbs 12486  TopSetcts 12567   ↾_t crest 12716  TopOpenctopn 12717

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-inn 8939  df-2 8997  df-3 8998  df-4 8999  df-5 9000  df-6 9001  df-7 9002  df-8 9003  df-9 9004  df-ndx 12489  df-slot 12490  df-base 12492  df-tset 12580  df-rest 12718  df-topn 12719

This theorem is referenced by:  sratopng  13730