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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 21567 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) ↔ ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦))) | |
2 | simpr 488 | . . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → 𝑥 = ∪ 𝑦) | |
3 | uniopn 21502 | . . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∪ 𝑦 ∈ 𝐽) | |
4 | 3 | adantr 484 | . . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → ∪ 𝑦 ∈ 𝐽) |
5 | 2, 4 | eqeltrd 2890 | . . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽) |
6 | 5 | expl 461 | . . . . 5 ⊢ (𝐽 ∈ Top → ((𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽)) |
7 | 6 | exlimdv 1934 | . . . 4 ⊢ (𝐽 ∈ Top → (∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽)) |
8 | 1, 7 | sylbid 243 | . . 3 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) → 𝑥 ∈ 𝐽)) |
9 | 8 | ssrdv 3921 | . 2 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) ⊆ 𝐽) |
10 | bastg 21571 | . 2 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (topGen‘𝐽)) | |
11 | 9, 10 | eqssd 3932 | 1 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ⊆ wss 3881 ∪ cuni 4800 ‘cfv 6324 topGenctg 16703 Topctop 21498 |
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 df-top 21499 |
This theorem is referenced by: eltop 21579 eltop2 21580 eltop3 21581 bastop 21586 tgtop11 21587 basgen 21593 tgfiss 21596 bastop1 21598 resttop 21765 dis1stc 22104 alexsubALTlem1 22652 xrtgioo 23411 topfne 33815 topfneec 33816 topfneec2 33817 dissneqlem 34757 |
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