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Mirrors > Home > MPE Home > Th. List > topontopn | Structured version Visualization version GIF version |
Description: Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
tsettps.a | ⊢ 𝐴 = (Base‘𝐾) |
tsettps.j | ⊢ 𝐽 = (TopSet‘𝐾) |
Ref | Expression |
---|---|
topontopn | ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponuni 22936 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 = ∪ 𝐽) | |
2 | eqimss2 4055 | . . . 4 ⊢ (𝐴 = ∪ 𝐽 → ∪ 𝐽 ⊆ 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → ∪ 𝐽 ⊆ 𝐴) |
4 | sspwuni 5105 | . . 3 ⊢ (𝐽 ⊆ 𝒫 𝐴 ↔ ∪ 𝐽 ⊆ 𝐴) | |
5 | 3, 4 | sylibr 234 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ⊆ 𝒫 𝐴) |
6 | tsettps.a | . . 3 ⊢ 𝐴 = (Base‘𝐾) | |
7 | tsettps.j | . . 3 ⊢ 𝐽 = (TopSet‘𝐾) | |
8 | 6, 7 | topnid 17482 | . 2 ⊢ (𝐽 ⊆ 𝒫 𝐴 → 𝐽 = (TopOpen‘𝐾)) |
9 | 5, 8 | syl 17 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 ‘cfv 6563 Basecbs 17245 TopSetcts 17304 TopOpenctopn 17468 TopOnctopon 22932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-rest 17469 df-topn 17470 df-topon 22933 |
This theorem is referenced by: tsettps 22963 xrstopn 23232 cnfldms 24812 cnfldtopn 24818 |
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