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Theorem bastop1 16981
Description: A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom " ( topGen `  B
)  =  J " to express " B is a basis for topology  J," since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
bastop1  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( ( topGen `  B
)  =  J  <->  A. x  e.  J  E. y
( y  C_  B  /\  x  =  U. y ) ) )
Distinct variable groups:    x, y, B    x, J, y

Proof of Theorem bastop1
StepHypRef Expression
1 tgss 16956 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  B )  C_  ( topGen `  J )
)
2 tgtop 16961 . . . . . 6  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
32adantr 452 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  J )  =  J )
41, 3sseqtrd 3327 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  B )  C_  J )
5 eqss 3306 . . . . 5  |-  ( (
topGen `  B )  =  J  <->  ( ( topGen `  B )  C_  J  /\  J  C_  ( topGen `  B ) ) )
65baib 872 . . . 4  |-  ( (
topGen `  B )  C_  J  ->  ( ( topGen `  B )  =  J  <-> 
J  C_  ( topGen `  B ) ) )
74, 6syl 16 . . 3  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( ( topGen `  B
)  =  J  <->  J  C_  ( topGen `
 B ) ) )
8 dfss3 3281 . . 3  |-  ( J 
C_  ( topGen `  B
)  <->  A. x  e.  J  x  e.  ( topGen `  B ) )
97, 8syl6bb 253 . 2  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( ( topGen `  B
)  =  J  <->  A. x  e.  J  x  e.  ( topGen `  B )
) )
10 ssexg 4290 . . . . 5  |-  ( ( B  C_  J  /\  J  e.  Top )  ->  B  e.  _V )
1110ancoms 440 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  J )  ->  B  e.  _V )
12 eltg3 16950 . . . 4  |-  ( B  e.  _V  ->  (
x  e.  ( topGen `  B )  <->  E. y
( y  C_  B  /\  x  =  U. y ) ) )
1311, 12syl 16 . . 3  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( x  e.  (
topGen `  B )  <->  E. y
( y  C_  B  /\  x  =  U. y ) ) )
1413ralbidv 2669 . 2  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( A. x  e.  J  x  e.  (
topGen `  B )  <->  A. x  e.  J  E. y
( y  C_  B  /\  x  =  U. y ) ) )
159, 14bitrd 245 1  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( ( topGen `  B
)  =  J  <->  A. x  e.  J  E. y
( y  C_  B  /\  x  =  U. y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899    C_ wss 3263   U.cuni 3957   ` cfv 5394   topGenctg 13592   Topctop 16881
This theorem is referenced by:  bastop2  16982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-topgen 13594  df-top 16886
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