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Mirrors > Home > ILE Home > Th. List > topnpropgd | GIF version |
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.) |
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) |
Ref | Expression |
---|---|
topnpropgd | ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) |
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 5724 | . 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿))) |
4 | topnpropgd.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
5 | eqid 2100 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | eqid 2100 | . . . 4 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
7 | 5, 6 | topnvalg 11914 | . . 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 2100 | . . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
11 | eqid 2100 | . . . 4 ⊢ (TopSet‘𝐿) = (TopSet‘𝐿) | |
12 | 10, 11 | topnvalg 11914 | . . 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 2140 | 1 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1299 ∈ wcel 1448 ‘cfv 5059 (class class class)co 5706 Basecbs 11741 TopSetcts 11809 ↾t crest 11902 TopOpenctopn 11903 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-cnex 7586 ax-resscn 7587 ax-1re 7589 ax-addrcl 7592 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-5 8640 df-6 8641 df-7 8642 df-8 8643 df-9 8644 df-ndx 11744 df-slot 11745 df-base 11747 df-tset 11822 df-rest 11904 df-topn 11905 |
This theorem is referenced by: (None) |
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