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Theorem bastop1 16731
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 16706 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  B )  C_  ( topGen `  J )
)
2 tgtop 16711 . . . . . 6  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
32adantr 451 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  J )  =  J )
41, 3sseqtrd 3214 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  B )  C_  J )
5 eqss 3194 . . . . 5  |-  ( (
topGen `  B )  =  J  <->  ( ( topGen `  B )  C_  J  /\  J  C_  ( topGen `  B ) ) )
65baib 871 . . . 4  |-  ( (
topGen `  B )  C_  J  ->  ( ( topGen `  B )  =  J  <-> 
J  C_  ( topGen `  B ) ) )
74, 6syl 15 . . 3  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( ( topGen `  B
)  =  J  <->  J  C_  ( topGen `
 B ) ) )
8 dfss3 3170 . . 3  |-  ( J 
C_  ( topGen `  B
)  <->  A. x  e.  J  x  e.  ( topGen `  B ) )
97, 8syl6bb 252 . 2  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( ( topGen `  B
)  =  J  <->  A. x  e.  J  x  e.  ( topGen `  B )
) )
10 ssexg 4160 . . . . 5  |-  ( ( B  C_  J  /\  J  e.  Top )  ->  B  e.  _V )
1110ancoms 439 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  J )  ->  B  e.  _V )
12 eltg3 16700 . . . 4  |-  ( B  e.  _V  ->  (
x  e.  ( topGen `  B )  <->  E. y
( y  C_  B  /\  x  =  U. y ) ) )
1311, 12syl 15 . . 3  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( x  e.  (
topGen `  B )  <->  E. y
( y  C_  B  /\  x  =  U. y ) ) )
1413ralbidv 2563 . 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 244 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 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   U.cuni 3827   ` cfv 5255   topGenctg 13342   Topctop 16631
This theorem is referenced by:  bastop2  16732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-topgen 13344  df-top 16636
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