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Mirrors > Home > ILE Home > Th. List > tsettps | GIF version |
Description: If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
tsettps.a | ⊢ 𝐴 = (Base‘𝐾) |
tsettps.j | ⊢ 𝐽 = (TopSet‘𝐾) |
Ref | Expression |
---|---|
tsettps | ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsettps.a | . . . 4 ⊢ 𝐴 = (Base‘𝐾) | |
2 | tsettps.j | . . . 4 ⊢ 𝐽 = (TopSet‘𝐾) | |
3 | 1, 2 | topontopn 12041 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾)) |
4 | id 19 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ (TopOn‘𝐴)) | |
5 | 3, 4 | eqeltrrd 2190 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (TopOpen‘𝐾) ∈ (TopOn‘𝐴)) |
6 | eqid 2113 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
7 | 1, 6 | istps 12036 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘𝐴)) |
8 | 5, 7 | sylibr 133 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1312 ∈ wcel 1461 ‘cfv 5079 Basecbs 11796 TopSetcts 11864 TopOpenctopn 11958 TopOnctopon 12014 TopSpctps 12034 |
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-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1re 7633 ax-addrcl 7636 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-inn 8625 df-2 8683 df-3 8684 df-4 8685 df-5 8686 df-6 8687 df-7 8688 df-8 8689 df-9 8690 df-ndx 11799 df-slot 11800 df-base 11802 df-tset 11877 df-rest 11959 df-topn 11960 df-top 12002 df-topon 12015 df-topsp 12035 |
This theorem is referenced by: eltpsg 12044 |
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