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Theorem eltg4i 12006
Description: An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
eltg4i  |-  ( A  e.  ( topGen `  B
)  ->  A  =  U. ( B  i^i  ~P A ) )

Proof of Theorem eltg4i
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-topgen 11923 . . . . . . 7  |-  topGen  =  ( x  e.  _V  |->  { y  |  y  C_  U. ( x  i^i  ~P y ) } )
21funmpt2 5098 . . . . . 6  |-  Fun  topGen
3 funrel 5076 . . . . . 6  |-  ( Fun  topGen  ->  Rel  topGen )
42, 3ax-mp 7 . . . . 5  |-  Rel  topGen
5 relelfvdm 5385 . . . . 5  |-  ( ( Rel  topGen  /\  A  e.  ( topGen `  B )
)  ->  B  e.  dom  topGen )
64, 5mpan 418 . . . 4  |-  ( A  e.  ( topGen `  B
)  ->  B  e.  dom  topGen )
7 eltg 12003 . . . 4  |-  ( B  e.  dom  topGen  ->  ( A  e.  ( topGen `  B )  <->  A  C_  U. ( B  i^i  ~P A ) ) )
86, 7syl 14 . . 3  |-  ( A  e.  ( topGen `  B
)  ->  ( A  e.  ( topGen `  B )  <->  A 
C_  U. ( B  i^i  ~P A ) ) )
98ibi 175 . 2  |-  ( A  e.  ( topGen `  B
)  ->  A  C_  U. ( B  i^i  ~P A ) )
10 inss2 3244 . . . . 5  |-  ( B  i^i  ~P A ) 
C_  ~P A
1110unissi 3706 . . . 4  |-  U. ( B  i^i  ~P A ) 
C_  U. ~P A
12 unipw 4077 . . . 4  |-  U. ~P A  =  A
1311, 12sseqtri 3081 . . 3  |-  U. ( B  i^i  ~P A ) 
C_  A
1413a1i 9 . 2  |-  ( A  e.  ( topGen `  B
)  ->  U. ( B  i^i  ~P A ) 
C_  A )
159, 14eqssd 3064 1  |-  ( A  e.  ( topGen `  B
)  ->  A  =  U. ( B  i^i  ~P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1299    e. wcel 1448   {cab 2086   _Vcvv 2641    i^i cin 3020    C_ wss 3021   ~Pcpw 3457   U.cuni 3683   dom cdm 4477   Rel wrel 4482   Fun wfun 5053   ` cfv 5059   topGenctg 11917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-topgen 11923
This theorem is referenced by:  eltg3  12008  tgdom  12023  tgidm  12025
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