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Theorem istopon 12217
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 funtopon 12216 . . . . 5  |-  Fun TopOn
2 funrel 5147 . . . . 5  |-  ( Fun TopOn  ->  Rel TopOn )
31, 2ax-mp 5 . . . 4  |-  Rel TopOn
4 relelfvdm 5460 . . . 4  |-  ( ( Rel TopOn  /\  J  e.  (TopOn `  B ) )  ->  B  e.  dom TopOn )
53, 4mpan 421 . . 3  |-  ( J  e.  (TopOn `  B
)  ->  B  e.  dom TopOn )
65elexd 2702 . 2  |-  ( J  e.  (TopOn `  B
)  ->  B  e.  _V )
7 uniexg 4368 . . . 4  |-  ( J  e.  Top  ->  U. J  e.  _V )
8 eleq1 2203 . . . 4  |-  ( B  =  U. J  -> 
( B  e.  _V  <->  U. J  e.  _V )
)
97, 8syl5ibrcom 156 . . 3  |-  ( J  e.  Top  ->  ( B  =  U. J  ->  B  e.  _V )
)
109imp 123 . 2  |-  ( ( J  e.  Top  /\  B  =  U. J )  ->  B  e.  _V )
11 eqeq1 2147 . . . . . 6  |-  ( b  =  B  ->  (
b  =  U. j  <->  B  =  U. j ) )
1211rabbidv 2678 . . . . 5  |-  ( b  =  B  ->  { j  e.  Top  |  b  =  U. j }  =  { j  e. 
Top  |  B  =  U. j } )
13 df-topon 12215 . . . . 5  |- TopOn  =  ( b  e.  _V  |->  { j  e.  Top  | 
b  =  U. j } )
14 vpwex 4110 . . . . . . 7  |-  ~P b  e.  _V
1514pwex 4114 . . . . . 6  |-  ~P ~P b  e.  _V
16 rabss 3178 . . . . . . 7  |-  ( { j  e.  Top  | 
b  =  U. j }  C_  ~P ~P b  <->  A. j  e.  Top  (
b  =  U. j  ->  j  e.  ~P ~P b ) )
17 pwuni 4123 . . . . . . . . . 10  |-  j  C_  ~P U. j
18 pweq 3517 . . . . . . . . . 10  |-  ( b  =  U. j  ->  ~P b  =  ~P U. j )
1917, 18sseqtrrid 3152 . . . . . . . . 9  |-  ( b  =  U. j  -> 
j  C_  ~P b
)
20 velpw 3521 . . . . . . . . 9  |-  ( j  e.  ~P ~P b  <->  j 
C_  ~P b )
2119, 20sylibr 133 . . . . . . . 8  |-  ( b  =  U. j  -> 
j  e.  ~P ~P b )
2221a1i 9 . . . . . . 7  |-  ( j  e.  Top  ->  (
b  =  U. j  ->  j  e.  ~P ~P b ) )
2316, 22mprgbir 2493 . . . . . 6  |-  { j  e.  Top  |  b  =  U. j } 
C_  ~P ~P b
2415, 23ssexi 4073 . . . . 5  |-  { j  e.  Top  |  b  =  U. j }  e.  _V
2512, 13, 24fvmpt3i 5508 . . . 4  |-  ( B  e.  _V  ->  (TopOn `  B )  =  {
j  e.  Top  |  B  =  U. j } )
2625eleq2d 2210 . . 3  |-  ( B  e.  _V  ->  ( J  e.  (TopOn `  B
)  <->  J  e.  { j  e.  Top  |  B  =  U. j } ) )
27 unieq 3752 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2827eqeq2d 2152 . . . 4  |-  ( j  =  J  ->  ( B  =  U. j  <->  B  =  U. J ) )
2928elrab 2843 . . 3  |-  ( J  e.  { j  e. 
Top  |  B  =  U. j }  <->  ( J  e.  Top  /\  B  = 
U. J ) )
3026, 29syl6bb 195 . 2  |-  ( B  e.  _V  ->  ( J  e.  (TopOn `  B
)  <->  ( J  e. 
Top  /\  B  =  U. J ) ) )
316, 10, 30pm5.21nii 694 1  |-  ( J  e.  (TopOn `  B
)  <->  ( J  e. 
Top  /\  B  =  U. J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   {crab 2421   _Vcvv 2689    C_ wss 3075   ~Pcpw 3514   U.cuni 3743   dom cdm 4546   Rel wrel 4551   Fun wfun 5124   ` cfv 5130   Topctop 12201  TopOnctopon 12214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-topon 12215
This theorem is referenced by:  topontop  12218  toponuni  12219  toptopon  12222  toponcom  12231  istps2  12237  tgtopon  12272  distopon  12293  epttop  12296  resttopon  12377  resttopon2  12384  txtopon  12468
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