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Theorem bastop1 16658
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 16633 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  B )  C_  ( topGen `  J )
)
2 tgtop 16638 . . . . . 6  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
32adantr 453 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  J )  =  J )
41, 3sseqtrd 3156 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  B )  C_  J )
5 eqss 3136 . . . . 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 3112 . . 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 4100 . . . . 5  |-  ( ( B  C_  J  /\  J  e.  Top )  ->  B  e.  _V )
1110ancoms 441 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  J )  ->  B  e.  _V )
12 eltg3 16627 . . . 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 2534 . 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 2516   _Vcvv 2740    C_ wss 3094   U.cuni 3768   ` cfv 4638   topGenctg 13269   Topctop 16558
This theorem is referenced by:  bastop2  16659
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 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-topgen 13271  df-top 16563
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