MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unitg Structured version   Visualization version   GIF version

Theorem unitg 22961
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 22958 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵)
2 velpw 4612 . . . . . 6 (𝑥 ∈ 𝒫 𝐵𝑥 𝐵)
31, 2sylibr 233 . . . . 5 (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 𝐵)
43ssriv 3983 . . . 4 (topGen‘𝐵) ⊆ 𝒫 𝐵
5 sspwuni 5108 . . . 4 ((topGen‘𝐵) ⊆ 𝒫 𝐵 (topGen‘𝐵) ⊆ 𝐵)
64, 5mpbi 229 . . 3 (topGen‘𝐵) ⊆ 𝐵
76a1i 11 . 2 (𝐵𝑉 (topGen‘𝐵) ⊆ 𝐵)
8 bastg 22960 . . 3 (𝐵𝑉𝐵 ⊆ (topGen‘𝐵))
98unissd 4923 . 2 (𝐵𝑉 𝐵 (topGen‘𝐵))
107, 9eqssd 3997 1 (𝐵𝑉 (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