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Mirrors > Home > ILE Home > Th. List > tgtop | 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 12215 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) ↔ ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦))) | |
2 | simpr 109 | . . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → 𝑥 = ∪ 𝑦) | |
3 | uniopn 12157 | . . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∪ 𝑦 ∈ 𝐽) | |
4 | 3 | adantr 274 | . . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → ∪ 𝑦 ∈ 𝐽) |
5 | 2, 4 | eqeltrd 2214 | . . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽) |
6 | 5 | expl 375 | . . . . 5 ⊢ (𝐽 ∈ Top → ((𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽)) |
7 | 6 | exlimdv 1791 | . . . 4 ⊢ (𝐽 ∈ Top → (∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽)) |
8 | 1, 7 | sylbid 149 | . . 3 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) → 𝑥 ∈ 𝐽)) |
9 | 8 | ssrdv 3098 | . 2 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) ⊆ 𝐽) |
10 | bastg 12219 | . 2 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (topGen‘𝐽)) | |
11 | 9, 10 | eqssd 3109 | 1 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ⊆ wss 3066 ∪ cuni 3731 ‘cfv 5118 topGenctg 12124 Topctop 12153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-topgen 12130 df-top 12154 |
This theorem is referenced by: eltop 12227 eltop2 12228 eltop3 12229 bastop 12233 tgtop11 12234 basgen 12238 bastop1 12241 resttop 12328 |
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