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Theorem unitg 12261
 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.)
Assertion
Ref Expression
unitg (𝐵𝑉 (topGen‘𝐵) = 𝐵)

Proof of Theorem unitg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tg1 12258 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵)
2 velpw 3518 . . . . . 6 (𝑥 ∈ 𝒫 𝐵𝑥 𝐵)
31, 2sylibr 133 . . . . 5 (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 𝐵)
43ssriv 3102 . . . 4 (topGen‘𝐵) ⊆ 𝒫 𝐵
5 sspwuni 3901 . . . 4 ((topGen‘𝐵) ⊆ 𝒫 𝐵 (topGen‘𝐵) ⊆ 𝐵)
64, 5mpbi 144 . . 3 (topGen‘𝐵) ⊆ 𝐵
76a1i 9 . 2 (𝐵𝑉 (topGen‘𝐵) ⊆ 𝐵)
8 bastg 12260 . . 3 (𝐵𝑉𝐵 ⊆ (topGen‘𝐵))
98unissd 3764 . 2 (𝐵𝑉 𝐵 (topGen‘𝐵))
107, 9eqssd 3115 1 (𝐵𝑉 (topGen‘𝐵) = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481   ⊆ wss 3072  𝒫 cpw 3511  ∪ cuni 3740  ‘cfv 5127  topGenctg 12165 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-iota 5092  df-fun 5129  df-fv 5135  df-topgen 12171 This theorem is referenced by:  tgcl  12263  tgtopon  12265  txtopon  12461  uniretop  12724
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