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Mirrors > Home > MPE Home > Th. List > unitg | Structured version Visualization version GIF version |
Description: The topology generated by a basis 𝐵 is a topology on ∪ 𝐵. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.) |
Ref | Expression |
---|---|
unitg | ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tg1 22265 | . . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵) | |
2 | velpw 4563 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐵 ↔ 𝑥 ⊆ ∪ 𝐵) | |
3 | 1, 2 | sylibr 233 | . . . . 5 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 ∪ 𝐵) |
4 | 3 | ssriv 3946 | . . . 4 ⊢ (topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵 |
5 | sspwuni 5058 | . . . 4 ⊢ ((topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵 ↔ ∪ (topGen‘𝐵) ⊆ ∪ 𝐵) | |
6 | 4, 5 | mpbi 229 | . . 3 ⊢ ∪ (topGen‘𝐵) ⊆ ∪ 𝐵 |
7 | 6 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) ⊆ ∪ 𝐵) |
8 | bastg 22267 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) | |
9 | 8 | unissd 4873 | . 2 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ⊆ ∪ (topGen‘𝐵)) |
10 | 7, 9 | eqssd 3959 | 1 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 𝒫 cpw 4558 ∪ cuni 4863 ‘cfv 6493 topGenctg 17278 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fv 6501 df-topgen 17284 |
This theorem is referenced by: tgcl 22270 tgtopon 22272 tgcmp 22703 2ndcsep 22761 txtopon 22893 ptuni 22896 xkouni 22901 prdstopn 22930 tgqtop 23014 alexsubb 23348 alexsubALTlem3 23351 alexsubALTlem4 23352 ptcmplem1 23354 uniretop 24077 fneval 34755 fnemeet1 34769 kelac2 41294 |
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