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Theorem toponsspwpwg 12199
Description: The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
Assertion
Ref Expression
toponsspwpwg  |-  ( A  e.  V  ->  (TopOn `  A )  C_  ~P ~P A )

Proof of Theorem toponsspwpwg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2697 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
2 rabssab 3184 . . . . . 6  |-  { y  e.  Top  |  A  =  U. y }  C_  { y  |  A  = 
U. y }
3 eqcom 2141 . . . . . . 7  |-  ( A  =  U. y  <->  U. y  =  A )
43abbii 2255 . . . . . 6  |-  { y  |  A  =  U. y }  =  {
y  |  U. y  =  A }
52, 4sseqtri 3131 . . . . 5  |-  { y  e.  Top  |  A  =  U. y }  C_  { y  |  U. y  =  A }
6 pwpwssunieq 3901 . . . . 5  |-  { y  |  U. y  =  A }  C_  ~P ~P A
75, 6sstri 3106 . . . 4  |-  { y  e.  Top  |  A  =  U. y }  C_  ~P ~P A
8 pwexg 4104 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
98pwexd 4105 . . . 4  |-  ( A  e.  V  ->  ~P ~P A  e.  _V )
10 ssexg 4067 . . . 4  |-  ( ( { y  e.  Top  |  A  =  U. y }  C_  ~P ~P A  /\  ~P ~P A  e. 
_V )  ->  { y  e.  Top  |  A  =  U. y }  e.  _V )
117, 9, 10sylancr 410 . . 3  |-  ( A  e.  V  ->  { y  e.  Top  |  A  =  U. y }  e.  _V )
12 eqeq1 2146 . . . . 5  |-  ( x  =  A  ->  (
x  =  U. y  <->  A  =  U. y ) )
1312rabbidv 2675 . . . 4  |-  ( x  =  A  ->  { y  e.  Top  |  x  =  U. y }  =  { y  e. 
Top  |  A  =  U. y } )
14 df-topon 12188 . . . 4  |- TopOn  =  ( x  e.  _V  |->  { y  e.  Top  |  x  =  U. y } )
1513, 14fvmptg 5497 . . 3  |-  ( ( A  e.  _V  /\  { y  e.  Top  |  A  =  U. y }  e.  _V )  ->  (TopOn `  A )  =  { y  e.  Top  |  A  =  U. y } )
161, 11, 15syl2anc 408 . 2  |-  ( A  e.  V  ->  (TopOn `  A )  =  {
y  e.  Top  |  A  =  U. y } )
1716, 7eqsstrdi 3149 1  |-  ( A  e.  V  ->  (TopOn `  A )  C_  ~P ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   {cab 2125   {crab 2420   _Vcvv 2686    C_ wss 3071   ~Pcpw 3510   U.cuni 3736   ` cfv 5123   Topctop 12174  TopOnctopon 12187
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-topon 12188
This theorem is referenced by: (None)
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