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Theorem unitg 21578
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 21575 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵)
2 velpw 4527 . . . . . 6 (𝑥 ∈ 𝒫 𝐵𝑥 𝐵)
31, 2sylibr 237 . . . . 5 (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 𝐵)
43ssriv 3957 . . . 4 (topGen‘𝐵) ⊆ 𝒫 𝐵
5 sspwuni 5008 . . . 4 ((topGen‘𝐵) ⊆ 𝒫 𝐵 (topGen‘𝐵) ⊆ 𝐵)
64, 5mpbi 233 . . 3 (topGen‘𝐵) ⊆ 𝐵
76a1i 11 . 2 (𝐵𝑉 (topGen‘𝐵) ⊆ 𝐵)
8 bastg 21577 . . 3 (𝐵𝑉𝐵 ⊆ (topGen‘𝐵))
98unissd 4834 . 2 (𝐵𝑉 𝐵 (topGen‘𝐵))
107, 9eqssd 3970 1 (𝐵𝑉 (topGen‘𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wss 3919  𝒫 cpw 4522   cuni 4824  cfv 6343  topGenctg 16711
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-topgen 16717
This theorem is referenced by:  tgcl  21580  tgtopon  21582  tgcmp  22012  2ndcsep  22070  txtopon  22202  ptuni  22205  xkouni  22210  prdstopn  22239  tgqtop  22323  alexsubb  22657  alexsubALTlem3  22660  alexsubALTlem4  22661  ptcmplem1  22663  uniretop  23374  fneval  33760  fnemeet1  33774  kelac2  39929
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