|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > tgtop | Structured version Visualization version GIF version | ||
| Description: A topology is its own basis. (Contributed by NM, 18-Jul-2006.) | 
| Ref | Expression | 
|---|---|
| tgtop | ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eltg3 22970 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) ↔ ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦))) | |
| 2 | simpr 484 | . . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → 𝑥 = ∪ 𝑦) | |
| 3 | uniopn 22904 | . . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∪ 𝑦 ∈ 𝐽) | |
| 4 | 3 | adantr 480 | . . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → ∪ 𝑦 ∈ 𝐽) | 
| 5 | 2, 4 | eqeltrd 2840 | . . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽) | 
| 6 | 5 | expl 457 | . . . . 5 ⊢ (𝐽 ∈ Top → ((𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽)) | 
| 7 | 6 | exlimdv 1932 | . . . 4 ⊢ (𝐽 ∈ Top → (∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽)) | 
| 8 | 1, 7 | sylbid 240 | . . 3 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) → 𝑥 ∈ 𝐽)) | 
| 9 | 8 | ssrdv 3988 | . 2 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) ⊆ 𝐽) | 
| 10 | bastg 22974 | . 2 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (topGen‘𝐽)) | |
| 11 | 9, 10 | eqssd 4000 | 1 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ⊆ wss 3950 ∪ cuni 4906 ‘cfv 6560 topGenctg 17483 Topctop 22900 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-topgen 17489 df-top 22901 | 
| This theorem is referenced by: eltop 22982 eltop2 22983 eltop3 22984 bastop 22989 tgtop11 22990 basgen 22996 tgfiss 22999 bastop1 23001 resttop 23169 dis1stc 23508 alexsubALTlem1 24056 xrtgioo 24829 topfne 36356 topfneec 36357 topfneec2 36358 dissneqlem 37342 | 
| Copyright terms: Public domain | W3C validator |