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Mirrors > Home > MPE Home > Th. List > tgclb | Structured version Visualization version GIF version |
Description: The property tgcl 21571 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 21571 | . 2 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top) | |
2 | 0opn 21506 | . . . . . . . . . 10 ⊢ ((topGen‘𝐵) ∈ Top → ∅ ∈ (topGen‘𝐵)) | |
3 | 2 | elfvexd 6699 | . . . . . . . . 9 ⊢ ((topGen‘𝐵) ∈ Top → 𝐵 ∈ V) |
4 | bastg 21568 | . . . . . . . . 9 ⊢ (𝐵 ∈ V → 𝐵 ⊆ (topGen‘𝐵)) | |
5 | 3, 4 | syl 17 | . . . . . . . 8 ⊢ ((topGen‘𝐵) ∈ Top → 𝐵 ⊆ (topGen‘𝐵)) |
6 | 5 | sselda 3967 | . . . . . . 7 ⊢ (((topGen‘𝐵) ∈ Top ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (topGen‘𝐵)) |
7 | 5 | sselda 3967 | . . . . . . 7 ⊢ (((topGen‘𝐵) ∈ Top ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (topGen‘𝐵)) |
8 | 6, 7 | anim12dan 620 | . . . . . 6 ⊢ (((topGen‘𝐵) ∈ Top ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵))) |
9 | inopn 21501 | . . . . . . 7 ⊢ (((topGen‘𝐵) ∈ Top ∧ 𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → (𝑥 ∩ 𝑦) ∈ (topGen‘𝐵)) | |
10 | 9 | 3expb 1116 | . . . . . 6 ⊢ (((topGen‘𝐵) ∈ Top ∧ (𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵))) → (𝑥 ∩ 𝑦) ∈ (topGen‘𝐵)) |
11 | 8, 10 | syldan 593 | . . . . 5 ⊢ (((topGen‘𝐵) ∈ Top ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∩ 𝑦) ∈ (topGen‘𝐵)) |
12 | tg2 21567 | . . . . . 6 ⊢ (((𝑥 ∩ 𝑦) ∈ (topGen‘𝐵) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))) | |
13 | 12 | ralrimiva 3182 | . . . . 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 21550 | . . . 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 259 | . 2 ⊢ ((topGen‘𝐵) ∈ Top → 𝐵 ∈ TopBases) |
19 | 1, 18 | impbii 211 | 1 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 Vcvv 3495 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 ‘cfv 6350 topGenctg 16705 Topctop 21495 TopBasesctb 21547 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-topgen 16711 df-top 21496 df-bases 21548 |
This theorem is referenced by: bastop2 21596 iocpnfordt 21817 icomnfordt 21818 iooordt 21819 tgcn 21854 tgcnp 21855 2ndcctbss 22057 2ndcomap 22060 dis2ndc 22062 flftg 22598 met2ndci 23126 xrtgioo 23408 topfneec 33698 |
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