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Theorem eltg4i 16915
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 5661 . . . 4  |-  ( A  e.  ( topGen `  B
)  ->  B  e.  dom  topGen )
2 eltg 16912 . . . 4  |-  ( B  e.  dom  topGen  ->  ( A  e.  ( topGen `  B )  <->  A  C_  U. ( B  i^i  ~P A ) ) )
31, 2syl 15 . . 3  |-  ( A  e.  ( topGen `  B
)  ->  ( A  e.  ( topGen `  B )  <->  A 
C_  U. ( B  i^i  ~P A ) ) )
43ibi 232 . 2  |-  ( A  e.  ( topGen `  B
)  ->  A  C_  U. ( B  i^i  ~P A ) )
5 inss2 3478 . . . . 5  |-  ( B  i^i  ~P A ) 
C_  ~P A
6 uniss 3950 . . . . 5  |-  ( ( B  i^i  ~P A
)  C_  ~P A  ->  U. ( B  i^i  ~P A )  C_  U. ~P A )
75, 6ax-mp 8 . . . 4  |-  U. ( B  i^i  ~P A ) 
C_  U. ~P A
8 unipw 4327 . . . 4  |-  U. ~P A  =  A
97, 8sseqtri 3296 . . 3  |-  U. ( B  i^i  ~P A ) 
C_  A
109a1i 10 . 2  |-  ( A  e.  ( topGen `  B
)  ->  U. ( B  i^i  ~P A ) 
C_  A )
114, 10eqssd 3282 1  |-  ( A  e.  ( topGen `  B
)  ->  A  =  U. ( B  i^i  ~P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1647    e. wcel 1715    i^i cin 3237    C_ wss 3238   ~Pcpw 3714   U.cuni 3929   dom cdm 4792   ` cfv 5358   topGenctg 13552
This theorem is referenced by:  eltg3  16917  tgdom  16933  tgidm  16935  ontgval  25697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-topgen 13554
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