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Theorem eltg4i 19221
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 5883 . . . 4  |-  ( A  e.  ( topGen `  B
)  ->  B  e.  dom  topGen )
2 eltg 19218 . . . 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 241 . 2  |-  ( A  e.  ( topGen `  B
)  ->  A  C_  U. ( B  i^i  ~P A ) )
5 inss2 3712 . . . . 5  |-  ( B  i^i  ~P A ) 
C_  ~P A
65unissi 4261 . . . 4  |-  U. ( B  i^i  ~P A ) 
C_  U. ~P A
7 unipw 4690 . . . 4  |-  U. ~P A  =  A
86, 7sseqtri 3529 . . 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 3514 1  |-  ( A  e.  ( topGen `  B
)  ->  A  =  U. ( B  i^i  ~P A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374    e. wcel 1762    i^i cin 3468    C_ wss 3469   ~Pcpw 4003   U.cuni 4238   dom cdm 4992   ` cfv 5579   topGenctg 14682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-topgen 14688
This theorem is referenced by:  eltg3  19223  tgdom  19239  tgidm  19241  ontgval  29459
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