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Theorem bastop1 17021
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 16996 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  B )  C_  ( topGen `  J )
)
2 tgtop 17001 . . . . . 6  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
32adantr 452 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  J )  =  J )
41, 3sseqtrd 3352 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  B )  C_  J )
5 eqss 3331 . . . . 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 3306 . . 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 4317 . . . . 5  |-  ( ( B  C_  J  /\  J  e.  Top )  ->  B  e.  _V )
1110ancoms 440 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  J )  ->  B  e.  _V )
12 eltg3 16990 . . . 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 2694 . 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 1721   A.wral 2674   _Vcvv 2924    C_ wss 3288   U.cuni 3983   ` cfv 5421   topGenctg 13628   Topctop 16921
This theorem is referenced by:  bastop2  17022
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-topgen 13630  df-top 16926
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