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Mirrors > Home > MPE Home > Th. List > tgclb | Structured version Visualization version GIF version |
Description: The property tgcl 21151 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 21151 | . 2 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top) | |
2 | 0opn 21086 | . . . . . . . . . 10 ⊢ ((topGen‘𝐵) ∈ Top → ∅ ∈ (topGen‘𝐵)) | |
3 | 2 | elfvexd 6472 | . . . . . . . . 9 ⊢ ((topGen‘𝐵) ∈ Top → 𝐵 ∈ V) |
4 | bastg 21148 | . . . . . . . . 9 ⊢ (𝐵 ∈ V → 𝐵 ⊆ (topGen‘𝐵)) | |
5 | 3, 4 | syl 17 | . . . . . . . 8 ⊢ ((topGen‘𝐵) ∈ Top → 𝐵 ⊆ (topGen‘𝐵)) |
6 | 5 | sselda 3827 | . . . . . . 7 ⊢ (((topGen‘𝐵) ∈ Top ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (topGen‘𝐵)) |
7 | 5 | sselda 3827 | . . . . . . 7 ⊢ (((topGen‘𝐵) ∈ Top ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (topGen‘𝐵)) |
8 | 6, 7 | anim12dan 612 | . . . . . 6 ⊢ (((topGen‘𝐵) ∈ Top ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵))) |
9 | inopn 21081 | . . . . . . 7 ⊢ (((topGen‘𝐵) ∈ Top ∧ 𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → (𝑥 ∩ 𝑦) ∈ (topGen‘𝐵)) | |
10 | 9 | 3expb 1153 | . . . . . 6 ⊢ (((topGen‘𝐵) ∈ Top ∧ (𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵))) → (𝑥 ∩ 𝑦) ∈ (topGen‘𝐵)) |
11 | 8, 10 | syldan 585 | . . . . 5 ⊢ (((topGen‘𝐵) ∈ Top ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∩ 𝑦) ∈ (topGen‘𝐵)) |
12 | tg2 21147 | . . . . . 6 ⊢ (((𝑥 ∩ 𝑦) ∈ (topGen‘𝐵) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))) | |
13 | 12 | ralrimiva 3175 | . . . . 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 3180 | . . 3 ⊢ ((topGen‘𝐵) ∈ Top → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))) |
16 | isbasis2g 21130 | . . . 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 249 | . 2 ⊢ ((topGen‘𝐵) ∈ Top → 𝐵 ∈ TopBases) |
19 | 1, 18 | impbii 201 | 1 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∈ wcel 2164 ∀wral 3117 ∃wrex 3118 Vcvv 3414 ∩ cin 3797 ⊆ wss 3798 ∅c0 4146 ‘cfv 6127 topGenctg 16458 Topctop 21075 TopBasesctb 21127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-iota 6090 df-fun 6129 df-fv 6135 df-topgen 16464 df-top 21076 df-bases 21128 |
This theorem is referenced by: bastop2 21176 iocpnfordt 21397 icomnfordt 21398 iooordt 21399 tgcn 21434 tgcnp 21435 2ndcctbss 21636 2ndcomap 21639 dis2ndc 21641 flftg 22177 met2ndci 22704 xrtgioo 22986 topfneec 32883 |
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