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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 21570 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) ↔ ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦))) | |
2 | simpr 487 | . . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → 𝑥 = ∪ 𝑦) | |
3 | uniopn 21505 | . . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∪ 𝑦 ∈ 𝐽) | |
4 | 3 | adantr 483 | . . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → ∪ 𝑦 ∈ 𝐽) |
5 | 2, 4 | eqeltrd 2913 | . . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽) |
6 | 5 | expl 460 | . . . . 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 242 | . . 3 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) → 𝑥 ∈ 𝐽)) |
9 | 8 | ssrdv 3973 | . 2 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) ⊆ 𝐽) |
10 | bastg 21574 | . 2 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (topGen‘𝐽)) | |
11 | 9, 10 | eqssd 3984 | 1 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ⊆ wss 3936 ∪ cuni 4838 ‘cfv 6355 topGenctg 16711 Topctop 21501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-topgen 16717 df-top 21502 |
This theorem is referenced by: eltop 21582 eltop2 21583 eltop3 21584 bastop 21589 tgtop11 21590 basgen 21596 tgfiss 21599 bastop1 21601 resttop 21768 dis1stc 22107 alexsubALTlem1 22655 xrtgioo 23414 topfne 33702 topfneec 33703 topfneec2 33704 dissneqlem 34624 |
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