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Theorem istopon 16983
Description: Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
istopon  |-  ( J  e.  (TopOn `  B
)  <->  ( J  e. 
Top  /\  B  =  U. J ) )

Proof of Theorem istopon
Dummy variables  b 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5751 . 2  |-  ( J  e.  (TopOn `  B
)  ->  B  e.  _V )
2 uniexg 4699 . . . 4  |-  ( J  e.  Top  ->  U. J  e.  _V )
3 eleq1 2496 . . . 4  |-  ( B  =  U. J  -> 
( B  e.  _V  <->  U. J  e.  _V )
)
42, 3syl5ibrcom 214 . . 3  |-  ( J  e.  Top  ->  ( B  =  U. J  ->  B  e.  _V )
)
54imp 419 . 2  |-  ( ( J  e.  Top  /\  B  =  U. J )  ->  B  e.  _V )
6 eqeq1 2442 . . . . . 6  |-  ( b  =  B  ->  (
b  =  U. j  <->  B  =  U. j ) )
76rabbidv 2941 . . . . 5  |-  ( b  =  B  ->  { j  e.  Top  |  b  =  U. j }  =  { j  e. 
Top  |  B  =  U. j } )
8 df-topon 16959 . . . . 5  |- TopOn  =  ( b  e.  _V  |->  { j  e.  Top  | 
b  =  U. j } )
9 vex 2952 . . . . . . . 8  |-  b  e. 
_V
109pwex 4375 . . . . . . 7  |-  ~P b  e.  _V
1110pwex 4375 . . . . . 6  |-  ~P ~P b  e.  _V
12 rabss 3413 . . . . . . 7  |-  ( { j  e.  Top  | 
b  =  U. j }  C_  ~P ~P b  <->  A. j  e.  Top  (
b  =  U. j  ->  j  e.  ~P ~P b ) )
13 pwuni 4388 . . . . . . . . . 10  |-  j  C_  ~P U. j
14 pweq 3795 . . . . . . . . . 10  |-  ( b  =  U. j  ->  ~P b  =  ~P U. j )
1513, 14syl5sseqr 3390 . . . . . . . . 9  |-  ( b  =  U. j  -> 
j  C_  ~P b
)
16 vex 2952 . . . . . . . . . 10  |-  j  e. 
_V
1716elpw 3798 . . . . . . . . 9  |-  ( j  e.  ~P ~P b  <->  j 
C_  ~P b )
1815, 17sylibr 204 . . . . . . . 8  |-  ( b  =  U. j  -> 
j  e.  ~P ~P b )
1918a1i 11 . . . . . . 7  |-  ( j  e.  Top  ->  (
b  =  U. j  ->  j  e.  ~P ~P b ) )
2012, 19mprgbir 2769 . . . . . 6  |-  { j  e.  Top  |  b  =  U. j } 
C_  ~P ~P b
2111, 20ssexi 4341 . . . . 5  |-  { j  e.  Top  |  b  =  U. j }  e.  _V
227, 8, 21fvmpt3i 5802 . . . 4  |-  ( B  e.  _V  ->  (TopOn `  B )  =  {
j  e.  Top  |  B  =  U. j } )
2322eleq2d 2503 . . 3  |-  ( B  e.  _V  ->  ( J  e.  (TopOn `  B
)  <->  J  e.  { j  e.  Top  |  B  =  U. j } ) )
24 unieq 4017 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2524eqeq2d 2447 . . . 4  |-  ( j  =  J  ->  ( B  =  U. j  <->  B  =  U. J ) )
2625elrab 3085 . . 3  |-  ( J  e.  { j  e. 
Top  |  B  =  U. j }  <->  ( J  e.  Top  /\  B  = 
U. J ) )
2723, 26syl6bb 253 . 2  |-  ( B  e.  _V  ->  ( J  e.  (TopOn `  B
)  <->  ( J  e. 
Top  /\  B  =  U. J ) ) )
281, 5, 27pm5.21nii 343 1  |-  ( J  e.  (TopOn `  B
)  <->  ( J  e. 
Top  /\  B  =  U. J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2702   _Vcvv 2949    C_ wss 3313   ~Pcpw 3792   U.cuni 4008   ` cfv 5447   Topctop 16951  TopOnctopon 16952
This theorem is referenced by:  topontop  16984  toponuni  16985  toponcom  16988  toptopon  16991  istps2  16995  tgtopon  17029  distopon  17054  indistopon  17058  fctop  17061  cctop  17063  ppttop  17064  epttop  17066  mretopd  17149  toponmre  17150  resttopon  17218  resttopon2  17225  kgentopon  17563  txtopon  17616  pttopon  17621  xkotopon  17625  qtoptopon  17729  flimtopon  17995  fclstopon  18037  fclsfnflim  18052  utoptopon  18259  onsuctopon  26177  neibastop1  26380  rfcnpre1  27658  cnfex  27667  stoweidlem47  27764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-topon 16959
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