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Theorem unitg 17024
 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 4011 . . . 4
2 eltg 17014 . . . . . 6
3 inss1 3553 . . . . . . . . 9
43unissi 4030 . . . . . . . 8
5 sstr 3348 . . . . . . . 8
64, 5mpan2 653 . . . . . . 7
76sseld 3339 . . . . . 6
82, 7syl6bi 220 . . . . 5
98rexlimdv 2821 . . . 4
101, 9syl5bi 209 . . 3
11 bastg 17023 . . . . 5
1211unissd 4031 . . . 4
1312sseld 3339 . . 3
1410, 13impbid 184 . 2
1514eqrdv 2433 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  wrex 2698   cin 3311   wss 3312  cpw 3791  cuni 4007  cfv 5446  ctg 13657 This theorem is referenced by:  tgcl  17026  tgtopon  17028  tgcmp  17456  2ndcsep  17514  txtopon  17615  ptuni  17618  xkouni  17623  prdstopn  17652  tgqtop  17736  alexsubb  18069  alexsubALTlem3  18072  alexsubALTlem4  18073  ptcmplem1  18075  uniretop  18788  fneval  26358  fnemeet1  26386  kelac2  27131 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-topgen 13659
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