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Theorem istopon 16995
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 5761 . 2  |-  ( J  e.  (TopOn `  B
)  ->  B  e.  _V )
2 uniexg 4709 . . . 4  |-  ( J  e.  Top  ->  U. J  e.  _V )
3 eleq1 2498 . . . 4  |-  ( B  =  U. J  -> 
( B  e.  _V  <->  U. J  e.  _V )
)
42, 3syl5ibrcom 215 . . 3  |-  ( J  e.  Top  ->  ( B  =  U. J  ->  B  e.  _V )
)
54imp 420 . 2  |-  ( ( J  e.  Top  /\  B  =  U. J )  ->  B  e.  _V )
6 eqeq1 2444 . . . . . 6  |-  ( b  =  B  ->  (
b  =  U. j  <->  B  =  U. j ) )
76rabbidv 2950 . . . . 5  |-  ( b  =  B  ->  { j  e.  Top  |  b  =  U. j }  =  { j  e. 
Top  |  B  =  U. j } )
8 df-topon 16971 . . . . 5  |- TopOn  =  ( b  e.  _V  |->  { j  e.  Top  | 
b  =  U. j } )
9 vex 2961 . . . . . . . 8  |-  b  e. 
_V
109pwex 4385 . . . . . . 7  |-  ~P b  e.  _V
1110pwex 4385 . . . . . 6  |-  ~P ~P b  e.  _V
12 rabss 3422 . . . . . . 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 4398 . . . . . . . . . 10  |-  j  C_  ~P U. j
14 pweq 3804 . . . . . . . . . 10  |-  ( b  =  U. j  ->  ~P b  =  ~P U. j )
1513, 14syl5sseqr 3399 . . . . . . . . 9  |-  ( b  =  U. j  -> 
j  C_  ~P b
)
16 vex 2961 . . . . . . . . . 10  |-  j  e. 
_V
1716elpw 3807 . . . . . . . . 9  |-  ( j  e.  ~P ~P b  <->  j 
C_  ~P b )
1815, 17sylibr 205 . . . . . . . 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 2778 . . . . . 6  |-  { j  e.  Top  |  b  =  U. j } 
C_  ~P ~P b
2111, 20ssexi 4351 . . . . 5  |-  { j  e.  Top  |  b  =  U. j }  e.  _V
227, 8, 21fvmpt3i 5812 . . . 4  |-  ( B  e.  _V  ->  (TopOn `  B )  =  {
j  e.  Top  |  B  =  U. j } )
2322eleq2d 2505 . . 3  |-  ( B  e.  _V  ->  ( J  e.  (TopOn `  B
)  <->  J  e.  { j  e.  Top  |  B  =  U. j } ) )
24 unieq 4026 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2524eqeq2d 2449 . . . 4  |-  ( j  =  J  ->  ( B  =  U. j  <->  B  =  U. J ) )
2625elrab 3094 . . 3  |-  ( J  e.  { j  e. 
Top  |  B  =  U. j }  <->  ( J  e.  Top  /\  B  = 
U. J ) )
2723, 26syl6bb 254 . 2  |-  ( B  e.  _V  ->  ( J  e.  (TopOn `  B
)  <->  ( J  e. 
Top  /\  B  =  U. J ) ) )
281, 5, 27pm5.21nii 344 1  |-  ( J  e.  (TopOn `  B
)  <->  ( J  e. 
Top  /\  B  =  U. J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   ` cfv 5457   Topctop 16963  TopOnctopon 16964
This theorem is referenced by:  topontop  16996  toponuni  16997  toponcom  17000  toptopon  17003  istps2  17007  tgtopon  17041  distopon  17066  indistopon  17070  fctop  17073  cctop  17075  ppttop  17076  epttop  17078  mretopd  17161  toponmre  17162  resttopon  17230  resttopon2  17237  kgentopon  17575  txtopon  17628  pttopon  17633  xkotopon  17637  qtoptopon  17741  flimtopon  18007  fclstopon  18049  fclsfnflim  18064  utoptopon  18271  onsuctopon  26189  neibastop1  26402  rfcnpre1  27680  cnfex  27689  stoweidlem47  27786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-topon 16971
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