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Mirrors > Home > MPE Home > Th. List > tgclb | Structured version Visualization version GIF version |
Description: The property tgcl 22958 can be reversed: if the topology generated by 𝐵 is actually a topology, then 𝐵 must be a topological basis. This yields an alternative definition of TopBases. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tgclb | ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcl 22958 | . 2 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top) | |
2 | 0opn 22892 | . . . . . . . . . 10 ⊢ ((topGen‘𝐵) ∈ Top → ∅ ∈ (topGen‘𝐵)) | |
3 | 2 | elfvexd 6930 | . . . . . . . . 9 ⊢ ((topGen‘𝐵) ∈ Top → 𝐵 ∈ V) |
4 | bastg 22955 | . . . . . . . . 9 ⊢ (𝐵 ∈ V → 𝐵 ⊆ (topGen‘𝐵)) | |
5 | 3, 4 | syl 17 | . . . . . . . 8 ⊢ ((topGen‘𝐵) ∈ Top → 𝐵 ⊆ (topGen‘𝐵)) |
6 | 5 | sselda 3979 | . . . . . . 7 ⊢ (((topGen‘𝐵) ∈ Top ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (topGen‘𝐵)) |
7 | 5 | sselda 3979 | . . . . . . 7 ⊢ (((topGen‘𝐵) ∈ Top ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (topGen‘𝐵)) |
8 | 6, 7 | anim12dan 617 | . . . . . 6 ⊢ (((topGen‘𝐵) ∈ Top ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵))) |
9 | inopn 22887 | . . . . . . 7 ⊢ (((topGen‘𝐵) ∈ Top ∧ 𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → (𝑥 ∩ 𝑦) ∈ (topGen‘𝐵)) | |
10 | 9 | 3expb 1117 | . . . . . 6 ⊢ (((topGen‘𝐵) ∈ Top ∧ (𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵))) → (𝑥 ∩ 𝑦) ∈ (topGen‘𝐵)) |
11 | 8, 10 | syldan 589 | . . . . 5 ⊢ (((topGen‘𝐵) ∈ Top ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∩ 𝑦) ∈ (topGen‘𝐵)) |
12 | tg2 22954 | . . . . . 6 ⊢ (((𝑥 ∩ 𝑦) ∈ (topGen‘𝐵) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))) | |
13 | 12 | ralrimiva 3136 | . . . . 5 ⊢ ((𝑥 ∩ 𝑦) ∈ (topGen‘𝐵) → ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))) |
14 | 11, 13 | syl 17 | . . . 4 ⊢ (((topGen‘𝐵) ∈ Top ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))) |
15 | 14 | ralrimivva 3191 | . . 3 ⊢ ((topGen‘𝐵) ∈ Top → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))) |
16 | isbasis2g 22937 | . . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))) | |
17 | 3, 16 | syl 17 | . . 3 ⊢ ((topGen‘𝐵) ∈ Top → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))) |
18 | 15, 17 | mpbird 256 | . 2 ⊢ ((topGen‘𝐵) ∈ Top → 𝐵 ∈ TopBases) |
19 | 1, 18 | impbii 208 | 1 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 Vcvv 3463 ∩ cin 3946 ⊆ wss 3947 ∅c0 4323 ‘cfv 6544 topGenctg 17445 Topctop 22881 TopBasesctb 22934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6496 df-fun 6546 df-fv 6552 df-topgen 17451 df-top 22882 df-bases 22935 |
This theorem is referenced by: bastop2 22983 iocpnfordt 23205 icomnfordt 23206 iooordt 23207 tgcn 23242 tgcnp 23243 2ndcctbss 23445 2ndcomap 23448 dis2ndc 23450 flftg 23986 met2ndci 24517 xrtgioo 24808 topfneec 36078 |
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