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Theorem eltg4i 17030
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
StepHypRef Expression
1 elfvdm 5760 . . . 4  |-  ( A  e.  ( topGen `  B
)  ->  B  e.  dom  topGen )
2 eltg 17027 . . . 4  |-  ( B  e.  dom  topGen  ->  ( A  e.  ( topGen `  B )  <->  A  C_  U. ( B  i^i  ~P A ) ) )
31, 2syl 16 . . 3  |-  ( A  e.  ( topGen `  B
)  ->  ( A  e.  ( topGen `  B )  <->  A 
C_  U. ( B  i^i  ~P A ) ) )
43ibi 234 . 2  |-  ( A  e.  ( topGen `  B
)  ->  A  C_  U. ( B  i^i  ~P A ) )
5 inss2 3564 . . . . 5  |-  ( B  i^i  ~P A ) 
C_  ~P A
65unissi 4040 . . . 4  |-  U. ( B  i^i  ~P A ) 
C_  U. ~P A
7 unipw 4417 . . . 4  |-  U. ~P A  =  A
86, 7sseqtri 3382 . . 3  |-  U. ( B  i^i  ~P A ) 
C_  A
98a1i 11 . 2  |-  ( A  e.  ( topGen `  B
)  ->  U. ( B  i^i  ~P A ) 
C_  A )
104, 9eqssd 3367 1  |-  ( A  e.  ( topGen `  B
)  ->  A  =  U. ( B  i^i  ~P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   dom cdm 4881   ` cfv 5457   topGenctg 13670
This theorem is referenced by:  eltg3  17032  tgdom  17048  tgidm  17050  ontgval  26186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-topgen 13672
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