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Theorem istopon 13171
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 13170 . . . . 5  |-  Fun TopOn
2 funrel 5229 . . . . 5  |-  ( Fun TopOn  ->  Rel TopOn )
31, 2ax-mp 5 . . . 4  |-  Rel TopOn
4 relelfvdm 5543 . . . 4  |-  ( ( Rel TopOn  /\  J  e.  (TopOn `  B ) )  ->  B  e.  dom TopOn )
53, 4mpan 424 . . 3  |-  ( J  e.  (TopOn `  B
)  ->  B  e.  dom TopOn )
65elexd 2750 . 2  |-  ( J  e.  (TopOn `  B
)  ->  B  e.  _V )
7 uniexg 4436 . . . 4  |-  ( J  e.  Top  ->  U. J  e.  _V )
8 eleq1 2240 . . . 4  |-  ( B  =  U. J  -> 
( B  e.  _V  <->  U. J  e.  _V )
)
97, 8syl5ibrcom 157 . . 3  |-  ( J  e.  Top  ->  ( B  =  U. J  ->  B  e.  _V )
)
109imp 124 . 2  |-  ( ( J  e.  Top  /\  B  =  U. J )  ->  B  e.  _V )
11 eqeq1 2184 . . . . . 6  |-  ( b  =  B  ->  (
b  =  U. j  <->  B  =  U. j ) )
1211rabbidv 2726 . . . . 5  |-  ( b  =  B  ->  { j  e.  Top  |  b  =  U. j }  =  { j  e. 
Top  |  B  =  U. j } )
13 df-topon 13169 . . . . 5  |- TopOn  =  ( b  e.  _V  |->  { j  e.  Top  | 
b  =  U. j } )
14 vpwex 4176 . . . . . . 7  |-  ~P b  e.  _V
1514pwex 4180 . . . . . 6  |-  ~P ~P b  e.  _V
16 rabss 3232 . . . . . . 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 4189 . . . . . . . . . 10  |-  j  C_  ~P U. j
18 pweq 3577 . . . . . . . . . 10  |-  ( b  =  U. j  ->  ~P b  =  ~P U. j )
1917, 18sseqtrrid 3206 . . . . . . . . 9  |-  ( b  =  U. j  -> 
j  C_  ~P b
)
20 velpw 3581 . . . . . . . . 9  |-  ( j  e.  ~P ~P b  <->  j 
C_  ~P b )
2119, 20sylibr 134 . . . . . . . 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 2535 . . . . . 6  |-  { j  e.  Top  |  b  =  U. j } 
C_  ~P ~P b
2415, 23ssexi 4138 . . . . 5  |-  { j  e.  Top  |  b  =  U. j }  e.  _V
2512, 13, 24fvmpt3i 5592 . . . 4  |-  ( B  e.  _V  ->  (TopOn `  B )  =  {
j  e.  Top  |  B  =  U. j } )
2625eleq2d 2247 . . 3  |-  ( B  e.  _V  ->  ( J  e.  (TopOn `  B
)  <->  J  e.  { j  e.  Top  |  B  =  U. j } ) )
27 unieq 3816 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2827eqeq2d 2189 . . . 4  |-  ( j  =  J  ->  ( B  =  U. j  <->  B  =  U. J ) )
2928elrab 2893 . . 3  |-  ( J  e.  { j  e. 
Top  |  B  =  U. j }  <->  ( J  e.  Top  /\  B  = 
U. J ) )
3026, 29bitrdi 196 . 2  |-  ( B  e.  _V  ->  ( J  e.  (TopOn `  B
)  <->  ( J  e. 
Top  /\  B  =  U. J ) ) )
316, 10, 30pm5.21nii 704 1  |-  ( J  e.  (TopOn `  B
)  <->  ( J  e. 
Top  /\  B  =  U. J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   {crab 2459   _Vcvv 2737    C_ wss 3129   ~Pcpw 3574   U.cuni 3807   dom cdm 4623   Rel wrel 4628   Fun wfun 5206   ` cfv 5212   Topctop 13155  TopOnctopon 13168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-topon 13169
This theorem is referenced by:  topontop  13172  toponuni  13173  toptopon  13176  toponcom  13185  istps2  13191  tgtopon  13226  distopon  13247  epttop  13250  resttopon  13331  resttopon2  13338  txtopon  13422
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