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Theorem eltg3 12153
Description: Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
Assertion
Ref Expression
eltg3  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  E. x
( x  C_  B  /\  A  =  U. x ) ) )
Distinct variable groups:    x, A    x, B    x, V

Proof of Theorem eltg3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-topgen 12068 . . . . . . 7  |-  topGen  =  ( x  e.  _V  |->  { y  |  y  C_  U. ( x  i^i  ~P y ) } )
21funmpt2 5132 . . . . . 6  |-  Fun  topGen
3 funrel 5110 . . . . . 6  |-  ( Fun  topGen  ->  Rel  topGen )
42, 3ax-mp 5 . . . . 5  |-  Rel  topGen
5 relelfvdm 5421 . . . . 5  |-  ( ( Rel  topGen  /\  A  e.  ( topGen `  B )
)  ->  B  e.  dom  topGen )
64, 5mpan 420 . . . 4  |-  ( A  e.  ( topGen `  B
)  ->  B  e.  dom  topGen )
7 inex1g 4034 . . . 4  |-  ( B  e.  dom  topGen  ->  ( B  i^i  ~P A )  e.  _V )
86, 7syl 14 . . 3  |-  ( A  e.  ( topGen `  B
)  ->  ( B  i^i  ~P A )  e. 
_V )
9 eltg4i 12151 . . 3  |-  ( A  e.  ( topGen `  B
)  ->  A  =  U. ( B  i^i  ~P A ) )
10 inss1 3266 . . . . . . 7  |-  ( B  i^i  ~P A ) 
C_  B
11 sseq1 3090 . . . . . . 7  |-  ( x  =  ( B  i^i  ~P A )  ->  (
x  C_  B  <->  ( B  i^i  ~P A )  C_  B ) )
1210, 11mpbiri 167 . . . . . 6  |-  ( x  =  ( B  i^i  ~P A )  ->  x  C_  B )
1312biantrurd 303 . . . . 5  |-  ( x  =  ( B  i^i  ~P A )  ->  ( A  =  U. x  <->  ( x  C_  B  /\  A  =  U. x
) ) )
14 unieq 3715 . . . . . 6  |-  ( x  =  ( B  i^i  ~P A )  ->  U. x  =  U. ( B  i^i  ~P A ) )
1514eqeq2d 2129 . . . . 5  |-  ( x  =  ( B  i^i  ~P A )  ->  ( A  =  U. x  <->  A  =  U. ( B  i^i  ~P A ) ) )
1613, 15bitr3d 189 . . . 4  |-  ( x  =  ( B  i^i  ~P A )  ->  (
( x  C_  B  /\  A  =  U. x )  <->  A  =  U. ( B  i^i  ~P A ) ) )
1716spcegv 2748 . . 3  |-  ( ( B  i^i  ~P A
)  e.  _V  ->  ( A  =  U. ( B  i^i  ~P A )  ->  E. x ( x 
C_  B  /\  A  =  U. x ) ) )
188, 9, 17sylc 62 . 2  |-  ( A  e.  ( topGen `  B
)  ->  E. x
( x  C_  B  /\  A  =  U. x ) )
19 eltg3i 12152 . . . . 5  |-  ( ( B  e.  V  /\  x  C_  B )  ->  U. x  e.  ( topGen `
 B ) )
20 eleq1 2180 . . . . 5  |-  ( A  =  U. x  -> 
( A  e.  (
topGen `  B )  <->  U. x  e.  ( topGen `  B )
) )
2119, 20syl5ibrcom 156 . . . 4  |-  ( ( B  e.  V  /\  x  C_  B )  -> 
( A  =  U. x  ->  A  e.  (
topGen `  B ) ) )
2221expimpd 360 . . 3  |-  ( B  e.  V  ->  (
( x  C_  B  /\  A  =  U. x )  ->  A  e.  ( topGen `  B )
) )
2322exlimdv 1775 . 2  |-  ( B  e.  V  ->  ( E. x ( x  C_  B  /\  A  =  U. x )  ->  A  e.  ( topGen `  B )
) )
2418, 23impbid2 142 1  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  E. x
( x  C_  B  /\  A  =  U. x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   E.wex 1453    e. wcel 1465   {cab 2103   _Vcvv 2660    i^i cin 3040    C_ wss 3041   ~Pcpw 3480   U.cuni 3706   dom cdm 4509   Rel wrel 4514   Fun wfun 5087   ` cfv 5093   topGenctg 12062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-topgen 12068
This theorem is referenced by:  tgval3  12154  tgtop  12164  eltop3  12167  tgidm  12170  bastop1  12179  tgrest  12265  tgcn  12304  txbasval  12363
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