MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istopon Structured version   Unicode version

Theorem istopon 18535
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 5722 . 2  |-  ( J  e.  (TopOn `  B
)  ->  B  e.  _V )
2 uniexg 6382 . . . 4  |-  ( J  e.  Top  ->  U. J  e.  _V )
3 eleq1 2503 . . . 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 2449 . . . . . 6  |-  ( b  =  B  ->  (
b  =  U. j  <->  B  =  U. j ) )
76rabbidv 2969 . . . . 5  |-  ( b  =  B  ->  { j  e.  Top  |  b  =  U. j }  =  { j  e. 
Top  |  B  =  U. j } )
8 df-topon 18511 . . . . 5  |- TopOn  =  ( b  e.  _V  |->  { j  e.  Top  | 
b  =  U. j } )
9 vex 2980 . . . . . . . 8  |-  b  e. 
_V
109pwex 4480 . . . . . . 7  |-  ~P b  e.  _V
1110pwex 4480 . . . . . 6  |-  ~P ~P b  e.  _V
12 rabss 3434 . . . . . . 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 4528 . . . . . . . . . 10  |-  j  C_  ~P U. j
14 pweq 3868 . . . . . . . . . 10  |-  ( b  =  U. j  ->  ~P b  =  ~P U. j )
1513, 14syl5sseqr 3410 . . . . . . . . 9  |-  ( b  =  U. j  -> 
j  C_  ~P b
)
16 selpw 3872 . . . . . . . . 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 2791 . . . . . 6  |-  { j  e.  Top  |  b  =  U. j } 
C_  ~P ~P b
2011, 19ssexi 4442 . . . . 5  |-  { j  e.  Top  |  b  =  U. j }  e.  _V
217, 8, 20fvmpt3i 5783 . . . 4  |-  ( B  e.  _V  ->  (TopOn `  B )  =  {
j  e.  Top  |  B  =  U. j } )
2221eleq2d 2510 . . 3  |-  ( B  e.  _V  ->  ( J  e.  (TopOn `  B
)  <->  J  e.  { j  e.  Top  |  B  =  U. j } ) )
23 unieq 4104 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2423eqeq2d 2454 . . . 4  |-  ( j  =  J  ->  ( B  =  U. j  <->  B  =  U. J ) )
2524elrab 3122 . . 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 1369    e. wcel 1756   {crab 2724   _Vcvv 2977    C_ wss 3333   ~Pcpw 3865   U.cuni 4096   ` cfv 5423   Topctop 18503  TopOnctopon 18504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-topon 18511
This theorem is referenced by:  topontop  18536  toponuni  18537  toponcom  18540  toptopon  18543  istps2  18547  tgtopon  18581  distopon  18606  indistopon  18610  fctop  18613  cctop  18615  ppttop  18616  epttop  18618  mretopd  18701  toponmre  18702  resttopon  18770  resttopon2  18777  kgentopon  19116  txtopon  19169  pttopon  19174  xkotopon  19178  qtoptopon  19282  flimtopon  19548  fclstopon  19590  fclsfnflim  19605  utoptopon  19816  onsuctopon  28285  neibastop1  28585  rfcnpre1  29746  cnfex  29755  stoweidlem47  29847
  Copyright terms: Public domain W3C validator