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Mirrors > Home > MPE Home > Th. List > bastg | Structured version Visualization version GIF version |
Description: A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
bastg | ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 478 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
2 | vex 3386 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
3 | 2 | pwid 4363 | . . . . . . 7 ⊢ 𝑥 ∈ 𝒫 𝑥 |
4 | 3 | a1i 11 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝒫 𝑥) |
5 | 1, 4 | elind 3994 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐵 ∩ 𝒫 𝑥)) |
6 | elssuni 4657 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)) |
8 | 7 | ex 402 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐵 → 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
9 | eltg 21087 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) | |
10 | 8, 9 | sylibrd 251 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐵 → 𝑥 ∈ (topGen‘𝐵))) |
11 | 10 | ssrdv 3802 | 1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ∩ cin 3766 ⊆ wss 3767 𝒫 cpw 4347 ∪ cuni 4626 ‘cfv 6099 topGenctg 16410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-iota 6062 df-fun 6101 df-fv 6107 df-topgen 16416 |
This theorem is referenced by: unitg 21097 tgclb 21100 tgtop 21103 tgidm 21110 tgss3 21116 bastop2 21124 elcls3 21213 ordtopn1 21324 ordtopn2 21325 leordtval2 21342 iocpnfordt 21345 icomnfordt 21346 iooordt 21347 tgcn 21382 tgcnp 21383 tgcmp 21530 2ndcsb 21578 2ndc1stc 21580 2ndcctbss 21584 2ndcomap 21587 ptopn 21712 xkoopn 21718 txopn 21731 txbasval 21735 ptpjcn 21740 flftg 22125 alexsubb 22175 blssopn 22625 iooretop 22894 bndth 23082 ovolicc2 23627 cncombf 23763 cnmbf 23764 ordtconnlem1 30478 elmbfmvol2 30837 dya2icoseg2 30848 iccllysconn 31741 rellysconn 31742 topjoin 32864 fnemeet2 32866 fnejoin1 32867 ontgval 32930 mblfinlem3 33929 mblfinlem4 33930 ismblfin 33931 cnambfre 33938 kelac2 38408 |
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