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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 488 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
2 | vex 3444 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
3 | 2 | pwid 4521 | . . . . . . 7 ⊢ 𝑥 ∈ 𝒫 𝑥 |
4 | 3 | a1i 11 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝒫 𝑥) |
5 | 1, 4 | elind 4121 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐵 ∩ 𝒫 𝑥)) |
6 | elssuni 4830 | . . . . 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 416 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐵 → 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
9 | eltg 21562 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) | |
10 | 8, 9 | sylibrd 262 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐵 → 𝑥 ∈ (topGen‘𝐵))) |
11 | 10 | ssrdv 3921 | 1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 ‘cfv 6324 topGenctg 16703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-topgen 16709 |
This theorem is referenced by: unitg 21572 tgclb 21575 tgtop 21578 tgidm 21585 tgss3 21591 bastop2 21599 elcls3 21688 ordtopn1 21799 ordtopn2 21800 leordtval2 21817 iocpnfordt 21820 icomnfordt 21821 iooordt 21822 tgcn 21857 tgcnp 21858 tgcmp 22006 2ndcsb 22054 2ndc1stc 22056 2ndcctbss 22060 2ndcomap 22063 ptopn 22188 xkoopn 22194 txopn 22207 txbasval 22211 ptpjcn 22216 flftg 22601 alexsubb 22651 blssopn 23102 iooretop 23371 bndth 23563 ovolicc2 24126 cncombf 24262 cnmbf 24263 ordtconnlem1 31277 elmbfmvol2 31635 dya2icoseg2 31646 iccllysconn 32610 rellysconn 32611 topjoin 33826 fnemeet2 33828 fnejoin1 33829 ontgval 33892 mblfinlem3 35096 mblfinlem4 35097 ismblfin 35098 cnambfre 35105 kelac2 40009 |
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