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Mirrors > Home > ILE Home > Th. List > unitg | 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 12600 | . . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵) | |
2 | velpw 3560 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐵 ↔ 𝑥 ⊆ ∪ 𝐵) | |
3 | 1, 2 | sylibr 133 | . . . . 5 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 ∪ 𝐵) |
4 | 3 | ssriv 3141 | . . . 4 ⊢ (topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵 |
5 | sspwuni 3944 | . . . 4 ⊢ ((topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵 ↔ ∪ (topGen‘𝐵) ⊆ ∪ 𝐵) | |
6 | 4, 5 | mpbi 144 | . . 3 ⊢ ∪ (topGen‘𝐵) ⊆ ∪ 𝐵 |
7 | 6 | a1i 9 | . 2 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) ⊆ ∪ 𝐵) |
8 | bastg 12602 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) | |
9 | 8 | unissd 3807 | . 2 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ⊆ ∪ (topGen‘𝐵)) |
10 | 7, 9 | eqssd 3154 | 1 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ⊆ wss 3111 𝒫 cpw 3553 ∪ cuni 3783 ‘cfv 5182 topGenctg 12507 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-topgen 12513 |
This theorem is referenced by: tgcl 12605 tgtopon 12607 txtopon 12803 uniretop 13066 |
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