![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 22958 | . . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵) | |
2 | velpw 4612 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐵 ↔ 𝑥 ⊆ ∪ 𝐵) | |
3 | 1, 2 | sylibr 233 | . . . . 5 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 ∪ 𝐵) |
4 | 3 | ssriv 3983 | . . . 4 ⊢ (topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵 |
5 | sspwuni 5108 | . . . 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 22960 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) | |
9 | 8 | unissd 4923 | . 2 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ⊆ ∪ (topGen‘𝐵)) |
10 | 7, 9 | eqssd 3997 | 1 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 𝒫 cpw 4607 ∪ cuni 4913 ‘cfv 6554 topGenctg 17452 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-topgen 17458 |
This theorem is referenced by: tgcl 22963 tgtopon 22965 tgcmp 23396 2ndcsep 23454 txtopon 23586 ptuni 23589 xkouni 23594 prdstopn 23623 tgqtop 23707 alexsubb 24041 alexsubALTlem3 24044 alexsubALTlem4 24045 ptcmplem1 24047 uniretop 24770 fneval 36064 fnemeet1 36078 kelac2 42726 |
Copyright terms: Public domain | W3C validator |