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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 487 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
2 | vex 3497 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
3 | 2 | pwid 4562 | . . . . . . 7 ⊢ 𝑥 ∈ 𝒫 𝑥 |
4 | 3 | a1i 11 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝒫 𝑥) |
5 | 1, 4 | elind 4170 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐵 ∩ 𝒫 𝑥)) |
6 | elssuni 4867 | . . . . 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 415 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐵 → 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
9 | eltg 21564 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) | |
10 | 8, 9 | sylibrd 261 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐵 → 𝑥 ∈ (topGen‘𝐵))) |
11 | 10 | ssrdv 3972 | 1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 ∪ cuni 4837 ‘cfv 6354 topGenctg 16710 |
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 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-topgen 16716 |
This theorem is referenced by: unitg 21574 tgclb 21577 tgtop 21580 tgidm 21587 tgss3 21593 bastop2 21601 elcls3 21690 ordtopn1 21801 ordtopn2 21802 leordtval2 21819 iocpnfordt 21822 icomnfordt 21823 iooordt 21824 tgcn 21859 tgcnp 21860 tgcmp 22008 2ndcsb 22056 2ndc1stc 22058 2ndcctbss 22062 2ndcomap 22065 ptopn 22190 xkoopn 22196 txopn 22209 txbasval 22213 ptpjcn 22218 flftg 22603 alexsubb 22653 blssopn 23104 iooretop 23373 bndth 23561 ovolicc2 24122 cncombf 24258 cnmbf 24259 ordtconnlem1 31167 elmbfmvol2 31525 dya2icoseg2 31536 iccllysconn 32497 rellysconn 32498 topjoin 33713 fnemeet2 33715 fnejoin1 33716 ontgval 33779 mblfinlem3 34930 mblfinlem4 34931 ismblfin 34932 cnambfre 34939 kelac2 39663 |
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