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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 13190 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) ↔ ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦))) | |
2 | simpr 110 | . . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → 𝑥 = ∪ 𝑦) | |
3 | uniopn 13132 | . . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∪ 𝑦 ∈ 𝐽) | |
4 | 3 | adantr 276 | . . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → ∪ 𝑦 ∈ 𝐽) |
5 | 2, 4 | eqeltrd 2254 | . . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽) |
6 | 5 | expl 378 | . . . . 5 ⊢ (𝐽 ∈ Top → ((𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽)) |
7 | 6 | exlimdv 1819 | . . . 4 ⊢ (𝐽 ∈ Top → (∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ 𝐽)) |
8 | 1, 7 | sylbid 150 | . . 3 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) → 𝑥 ∈ 𝐽)) |
9 | 8 | ssrdv 3161 | . 2 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) ⊆ 𝐽) |
10 | bastg 13194 | . 2 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (topGen‘𝐽)) | |
11 | 9, 10 | eqssd 3172 | 1 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ⊆ wss 3129 ∪ cuni 3807 ‘cfv 5211 topGenctg 12638 Topctop 13128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-topgen 12644 df-top 13129 |
This theorem is referenced by: eltop 13202 eltop2 13203 eltop3 13204 bastop 13208 tgtop11 13209 basgen 13213 bastop1 13216 resttop 13303 |
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