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Theorem unitg 22333
Description: The topology generated by a basis 𝐡 is a topology on βˆͺ 𝐡. 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 (𝐡 ∈ 𝑉 β†’ βˆͺ (topGenβ€˜π΅) = βˆͺ 𝐡)

Proof of Theorem unitg
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 tg1 22330 . . . . . 6 (π‘₯ ∈ (topGenβ€˜π΅) β†’ π‘₯ βŠ† βˆͺ 𝐡)
2 velpw 4566 . . . . . 6 (π‘₯ ∈ 𝒫 βˆͺ 𝐡 ↔ π‘₯ βŠ† βˆͺ 𝐡)
31, 2sylibr 233 . . . . 5 (π‘₯ ∈ (topGenβ€˜π΅) β†’ π‘₯ ∈ 𝒫 βˆͺ 𝐡)
43ssriv 3949 . . . 4 (topGenβ€˜π΅) βŠ† 𝒫 βˆͺ 𝐡
5 sspwuni 5061 . . . 4 ((topGenβ€˜π΅) βŠ† 𝒫 βˆͺ 𝐡 ↔ βˆͺ (topGenβ€˜π΅) βŠ† βˆͺ 𝐡)
64, 5mpbi 229 . . 3 βˆͺ (topGenβ€˜π΅) βŠ† βˆͺ 𝐡
76a1i 11 . 2 (𝐡 ∈ 𝑉 β†’ βˆͺ (topGenβ€˜π΅) βŠ† βˆͺ 𝐡)
8 bastg 22332 . . 3 (𝐡 ∈ 𝑉 β†’ 𝐡 βŠ† (topGenβ€˜π΅))
98unissd 4876 . 2 (𝐡 ∈ 𝑉 β†’ βˆͺ 𝐡 βŠ† βˆͺ (topGenβ€˜π΅))
107, 9eqssd 3962 1 (𝐡 ∈ 𝑉 β†’ βˆͺ (topGenβ€˜π΅) = βˆͺ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911  π’« cpw 4561  βˆͺ cuni 4866  β€˜cfv 6497  topGenctg 17324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-topgen 17330
This theorem is referenced by:  tgcl  22335  tgtopon  22337  tgcmp  22768  2ndcsep  22826  txtopon  22958  ptuni  22961  xkouni  22966  prdstopn  22995  tgqtop  23079  alexsubb  23413  alexsubALTlem3  23416  alexsubALTlem4  23417  ptcmplem1  23419  uniretop  24142  fneval  34870  fnemeet1  34884  kelac2  41435
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