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Theorem istopon 19221
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 5893 . 2  |-  ( J  e.  (TopOn `  B
)  ->  B  e.  _V )
2 uniexg 6581 . . . 4  |-  ( J  e.  Top  ->  U. J  e.  _V )
3 eleq1 2539 . . . 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 429 . 2  |-  ( ( J  e.  Top  /\  B  =  U. J )  ->  B  e.  _V )
6 eqeq1 2471 . . . . . 6  |-  ( b  =  B  ->  (
b  =  U. j  <->  B  =  U. j ) )
76rabbidv 3105 . . . . 5  |-  ( b  =  B  ->  { j  e.  Top  |  b  =  U. j }  =  { j  e. 
Top  |  B  =  U. j } )
8 df-topon 19197 . . . . 5  |- TopOn  =  ( b  e.  _V  |->  { j  e.  Top  | 
b  =  U. j } )
9 vex 3116 . . . . . . . 8  |-  b  e. 
_V
109pwex 4630 . . . . . . 7  |-  ~P b  e.  _V
1110pwex 4630 . . . . . 6  |-  ~P ~P b  e.  _V
12 rabss 3577 . . . . . . 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 4678 . . . . . . . . . 10  |-  j  C_  ~P U. j
14 pweq 4013 . . . . . . . . . 10  |-  ( b  =  U. j  ->  ~P b  =  ~P U. j )
1513, 14syl5sseqr 3553 . . . . . . . . 9  |-  ( b  =  U. j  -> 
j  C_  ~P b
)
16 selpw 4017 . . . . . . . . 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 2828 . . . . . 6  |-  { j  e.  Top  |  b  =  U. j } 
C_  ~P ~P b
2011, 19ssexi 4592 . . . . 5  |-  { j  e.  Top  |  b  =  U. j }  e.  _V
217, 8, 20fvmpt3i 5954 . . . 4  |-  ( B  e.  _V  ->  (TopOn `  B )  =  {
j  e.  Top  |  B  =  U. j } )
2221eleq2d 2537 . . 3  |-  ( B  e.  _V  ->  ( J  e.  (TopOn `  B
)  <->  J  e.  { j  e.  Top  |  B  =  U. j } ) )
23 unieq 4253 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2423eqeq2d 2481 . . . 4  |-  ( j  =  J  ->  ( B  =  U. j  <->  B  =  U. J ) )
2524elrab 3261 . . 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 353 1  |-  ( J  e.  (TopOn `  B
)  <->  ( J  e. 
Top  /\  B  =  U. J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   ` cfv 5588   Topctop 19189  TopOnctopon 19190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-topon 19197
This theorem is referenced by:  topontop  19222  toponuni  19223  toponcom  19226  toptopon  19229  istps2  19233  tgtopon  19267  distopon  19292  indistopon  19296  fctop  19299  cctop  19301  ppttop  19302  epttop  19304  mretopd  19387  toponmre  19388  resttopon  19456  resttopon2  19463  kgentopon  19802  txtopon  19855  pttopon  19860  xkotopon  19864  qtoptopon  19968  flimtopon  20234  fclstopon  20276  fclsfnflim  20291  utoptopon  20502  qtopt1  27664  onsuctopon  29504  neibastop1  29808  rfcnpre1  31000  cnfex  31009  islptre  31189  icccncfext  31254  stoweidlem47  31375
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