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Theorem unitg 13455
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 13452 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵)
2 velpw 3582 . . . . . 6 (𝑥 ∈ 𝒫 𝐵𝑥 𝐵)
31, 2sylibr 134 . . . . 5 (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 𝐵)
43ssriv 3159 . . . 4 (topGen‘𝐵) ⊆ 𝒫 𝐵
5 sspwuni 3971 . . . 4 ((topGen‘𝐵) ⊆ 𝒫 𝐵 (topGen‘𝐵) ⊆ 𝐵)
64, 5mpbi 145 . . 3 (topGen‘𝐵) ⊆ 𝐵
76a1i 9 . 2 (𝐵𝑉 (topGen‘𝐵) ⊆ 𝐵)
8 bastg 13454 . . 3 (𝐵𝑉𝐵 ⊆ (topGen‘𝐵))
98unissd 3833 . 2 (𝐵𝑉 𝐵 (topGen‘𝐵))
107, 9eqssd 3172 1 (𝐵𝑉 (topGen‘𝐵) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  wss 3129  𝒫 cpw 3575   cuni 3809  cfv 5216  topGenctg 12693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-topgen 12699
This theorem is referenced by:  tgcl  13457  tgtopon  13459  txtopon  13655  uniretop  13918
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