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Theorem unitg 20973
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 20970 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵)
2 selpw 4309 . . . . . 6 (𝑥 ∈ 𝒫 𝐵𝑥 𝐵)
31, 2sylibr 224 . . . . 5 (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 𝐵)
43ssriv 3748 . . . 4 (topGen‘𝐵) ⊆ 𝒫 𝐵
5 sspwuni 4763 . . . 4 ((topGen‘𝐵) ⊆ 𝒫 𝐵 (topGen‘𝐵) ⊆ 𝐵)
64, 5mpbi 220 . . 3 (topGen‘𝐵) ⊆ 𝐵
76a1i 11 . 2 (𝐵𝑉 (topGen‘𝐵) ⊆ 𝐵)
8 bastg 20972 . . 3 (𝐵𝑉𝐵 ⊆ (topGen‘𝐵))
98unissd 4614 . 2 (𝐵𝑉 𝐵 (topGen‘𝐵))
107, 9eqssd 3761 1 (𝐵𝑉 (topGen‘𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wss 3715  𝒫 cpw 4302   cuni 4588  cfv 6049  topGenctg 16300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-topgen 16306
This theorem is referenced by:  tgcl  20975  tgtopon  20977  tgcmp  21406  2ndcsep  21464  txtopon  21596  ptuni  21599  xkouni  21604  prdstopn  21633  tgqtop  21717  alexsubb  22051  alexsubALTlem3  22054  alexsubALTlem4  22055  ptcmplem1  22057  uniretop  22767  fneval  32653  fnemeet1  32667  kelac2  38137
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