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Theorem unitg 20819
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 20816 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵)
2 selpw 4198 . . . . . 6 (𝑥 ∈ 𝒫 𝐵𝑥 𝐵)
31, 2sylibr 224 . . . . 5 (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 𝐵)
43ssriv 3640 . . . 4 (topGen‘𝐵) ⊆ 𝒫 𝐵
5 sspwuni 4643 . . . 4 ((topGen‘𝐵) ⊆ 𝒫 𝐵 (topGen‘𝐵) ⊆ 𝐵)
64, 5mpbi 220 . . 3 (topGen‘𝐵) ⊆ 𝐵
76a1i 11 . 2 (𝐵𝑉 (topGen‘𝐵) ⊆ 𝐵)
8 bastg 20818 . . 3 (𝐵𝑉𝐵 ⊆ (topGen‘𝐵))
98unissd 4494 . 2 (𝐵𝑉 𝐵 (topGen‘𝐵))
107, 9eqssd 3653 1 (𝐵𝑉 (topGen‘𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wss 3607  𝒫 cpw 4191   cuni 4468  cfv 5926  topGenctg 16145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151
This theorem is referenced by:  tgcl  20821  tgtopon  20823  tgcmp  21252  2ndcsep  21310  txtopon  21442  ptuni  21445  xkouni  21450  prdstopn  21479  tgqtop  21563  alexsubb  21897  alexsubALTlem3  21900  alexsubALTlem4  21901  ptcmplem1  21903  uniretop  22613  fneval  32472  fnemeet1  32486  kelac2  37952
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