Theorem topnpropgd 13550

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 6076	. 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿)))
4		topnpropgd.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
5		eqid 2234	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		eqid 2234	. . . 4 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
7	5, 6	topnvalg 13548	. . 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 2234	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
11		eqid 2234	. . . 4 ⊢ (TopSet‘𝐿) = (TopSet‘𝐿)
12	10, 11	topnvalg 13548	. . 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 2275	1 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  ‘cfv 5357  (class class class)co 6058  Basecbs 13296  TopSetcts 13380   ↾_t crest 13536  TopOpenctopn 13537

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-ndx 13299  df-slot 13300  df-base 13302  df-tset 13393  df-rest 13538  df-topn 13539

This theorem is referenced by:  sratopng  14721