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Theorem bastop1 16693
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 16668 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  B )  C_  ( topGen `  J )
)
2 tgtop 16673 . . . . . 6  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
32adantr 453 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  J )  =  J )
41, 3sseqtrd 3189 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  B )  C_  J )
5 eqss 3169 . . . . 5  |-  ( (
topGen `  B )  =  J  <->  ( ( topGen `  B )  C_  J  /\  J  C_  ( topGen `  B ) ) )
65baib 876 . . . 4  |-  ( (
topGen `  B )  C_  J  ->  ( ( topGen `  B )  =  J  <-> 
J  C_  ( topGen `  B ) ) )
74, 6syl 17 . . 3  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( ( topGen `  B
)  =  J  <->  J  C_  ( topGen `
 B ) ) )
8 dfss3 3145 . . 3  |-  ( J 
C_  ( topGen `  B
)  <->  A. x  e.  J  x  e.  ( topGen `  B ) )
97, 8syl6bb 254 . 2  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( ( topGen `  B
)  =  J  <->  A. x  e.  J  x  e.  ( topGen `  B )
) )
10 ssexg 4134 . . . . 5  |-  ( ( B  C_  J  /\  J  e.  Top )  ->  B  e.  _V )
1110ancoms 441 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  J )  ->  B  e.  _V )
12 eltg3 16662 . . . 4  |-  ( B  e.  _V  ->  (
x  e.  ( topGen `  B )  <->  E. y
( y  C_  B  /\  x  =  U. y ) ) )
1311, 12syl 17 . . 3  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( x  e.  (
topGen `  B )  <->  E. y
( y  C_  B  /\  x  =  U. y ) ) )
1413ralbidv 2538 . 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 246 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 6    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   A.wral 2518   _Vcvv 2763    C_ wss 3127   U.cuni 3801   ` cfv 4673   topGenctg 13304   Topctop 16593
This theorem is referenced by:  bastop2  16694
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fv 4689  df-topgen 13306  df-top 16598
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