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

Theorem unitg 16705
 Description: The topology generated by a basis is a topology on . 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 completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.)
Assertion
Ref Expression
unitg

Proof of Theorem unitg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 3831 . . . 4
2 eltg 16695 . . . . . 6
3 inss1 3389 . . . . . . . . 9
4 uniss 3848 . . . . . . . . 9
53, 4ax-mp 8 . . . . . . . 8
6 sstr 3187 . . . . . . . 8
75, 6mpan2 652 . . . . . . 7
87sseld 3179 . . . . . 6
92, 8syl6bi 219 . . . . 5
109rexlimdv 2666 . . . 4
111, 10syl5bi 208 . . 3
12 bastg 16704 . . . . 5
13 uniss 3848 . . . . 5
1412, 13syl 15 . . . 4
1514sseld 3179 . . 3
1611, 15impbid 183 . 2
1716eqrdv 2281 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1623   wcel 1684  wrex 2544   cin 3151   wss 3152  cpw 3625  cuni 3827  cfv 5255  ctg 13342 This theorem is referenced by:  tgcl  16707  tgtopon  16709  tgcmp  17128  2ndcsep  17185  txtopon  17286  ptuni  17289  xkouni  17294  prdstopn  17322  tgqtop  17403  alexsubb  17740  alexsubALTlem3  17743  alexsubALTlem4  17744  ptcmplem1  17746  uniretop  18271  fneval  26287  fnemeet1  26315  kelac2  27163 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-topgen 13344
 Copyright terms: Public domain W3C validator