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Theorem istopon 19593
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 5875 . 2  |-  ( J  e.  (TopOn `  B
)  ->  B  e.  _V )
2 uniexg 6570 . . . 4  |-  ( J  e.  Top  ->  U. J  e.  _V )
3 eleq1 2526 . . . 4  |-  ( B  =  U. J  -> 
( B  e.  _V  <->  U. J  e.  _V )
)
42, 3syl5ibrcom 222 . . 3  |-  ( J  e.  Top  ->  ( B  =  U. J  ->  B  e.  _V )
)
54imp 427 . 2  |-  ( ( J  e.  Top  /\  B  =  U. J )  ->  B  e.  _V )
6 eqeq1 2458 . . . . . 6  |-  ( b  =  B  ->  (
b  =  U. j  <->  B  =  U. j ) )
76rabbidv 3098 . . . . 5  |-  ( b  =  B  ->  { j  e.  Top  |  b  =  U. j }  =  { j  e. 
Top  |  B  =  U. j } )
8 df-topon 19569 . . . . 5  |- TopOn  =  ( b  e.  _V  |->  { j  e.  Top  | 
b  =  U. j } )
9 vex 3109 . . . . . . . 8  |-  b  e. 
_V
109pwex 4620 . . . . . . 7  |-  ~P b  e.  _V
1110pwex 4620 . . . . . 6  |-  ~P ~P b  e.  _V
12 rabss 3563 . . . . . . 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 4668 . . . . . . . . . 10  |-  j  C_  ~P U. j
14 pweq 4002 . . . . . . . . . 10  |-  ( b  =  U. j  ->  ~P b  =  ~P U. j )
1513, 14syl5sseqr 3538 . . . . . . . . 9  |-  ( b  =  U. j  -> 
j  C_  ~P b
)
16 selpw 4006 . . . . . . . . 9  |-  ( j  e.  ~P ~P b  <->  j 
C_  ~P b )
1715, 16sylibr 212 . . . . . . . 8  |-  ( b  =  U. j  -> 
j  e.  ~P ~P b )
1817a1i 11 . . . . . . 7  |-  ( j  e.  Top  ->  (
b  =  U. j  ->  j  e.  ~P ~P b ) )
1912, 18mprgbir 2818 . . . . . 6  |-  { j  e.  Top  |  b  =  U. j } 
C_  ~P ~P b
2011, 19ssexi 4582 . . . . 5  |-  { j  e.  Top  |  b  =  U. j }  e.  _V
217, 8, 20fvmpt3i 5935 . . . 4  |-  ( B  e.  _V  ->  (TopOn `  B )  =  {
j  e.  Top  |  B  =  U. j } )
2221eleq2d 2524 . . 3  |-  ( B  e.  _V  ->  ( J  e.  (TopOn `  B
)  <->  J  e.  { j  e.  Top  |  B  =  U. j } ) )
23 unieq 4243 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2423eqeq2d 2468 . . . 4  |-  ( j  =  J  ->  ( B  =  U. j  <->  B  =  U. J ) )
2524elrab 3254 . . 3  |-  ( J  e.  { j  e. 
Top  |  B  =  U. j }  <->  ( J  e.  Top  /\  B  = 
U. J ) )
2622, 25syl6bb 261 . 2  |-  ( B  e.  _V  ->  ( J  e.  (TopOn `  B
)  <->  ( J  e. 
Top  /\  B  =  U. J ) ) )
271, 5, 26pm5.21nii 351 1  |-  ( J  e.  (TopOn `  B
)  <->  ( J  e. 
Top  /\  B  =  U. J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   ` cfv 5570   Topctop 19561  TopOnctopon 19562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-topon 19569
This theorem is referenced by:  topontop  19594  toponuni  19595  toponcom  19598  toptopon  19601  istps2  19605  tgtopon  19640  distopon  19665  indistopon  19669  fctop  19672  cctop  19674  ppttop  19675  epttop  19677  mretopd  19760  toponmre  19761  resttopon  19829  resttopon2  19836  kgentopon  20205  txtopon  20258  pttopon  20263  xkotopon  20267  qtoptopon  20371  flimtopon  20637  fclstopon  20679  fclsfnflim  20694  utoptopon  20905  qtopt1  28073  onsuctopon  30127  neibastop1  30417  rfcnpre1  31634  cnfex  31643  icccncfext  31929  stoweidlem47  32068
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