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Theorem unitg 20677
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 20674 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵)
2 selpw 4142 . . . . . 6 (𝑥 ∈ 𝒫 𝐵𝑥 𝐵)
31, 2sylibr 224 . . . . 5 (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 𝐵)
43ssriv 3592 . . . 4 (topGen‘𝐵) ⊆ 𝒫 𝐵
5 sspwuni 4582 . . . 4 ((topGen‘𝐵) ⊆ 𝒫 𝐵 (topGen‘𝐵) ⊆ 𝐵)
64, 5mpbi 220 . . 3 (topGen‘𝐵) ⊆ 𝐵
76a1i 11 . 2 (𝐵𝑉 (topGen‘𝐵) ⊆ 𝐵)
8 bastg 20676 . . 3 (𝐵𝑉𝐵 ⊆ (topGen‘𝐵))
98unissd 4433 . 2 (𝐵𝑉 𝐵 (topGen‘𝐵))
107, 9eqssd 3605 1 (𝐵𝑉 (topGen‘𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  wss 3560  𝒫 cpw 4135   cuni 4407  cfv 5850  topGenctg 16014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-topgen 16020
This theorem is referenced by:  tgcl  20679  tgtopon  20681  tgcmp  21109  2ndcsep  21167  txtopon  21299  ptuni  21302  xkouni  21307  prdstopn  21336  tgqtop  21420  alexsubb  21755  alexsubALTlem3  21758  alexsubALTlem4  21759  ptcmplem1  21761  uniretop  22471  fneval  31981  fnemeet1  31995  kelac2  37101
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