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Theorem toponsspwpwg 13926
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 2763 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
2 rabssab 3258 . . . . . 6  |-  { y  e.  Top  |  A  =  U. y }  C_  { y  |  A  = 
U. y }
3 eqcom 2191 . . . . . . 7  |-  ( A  =  U. y  <->  U. y  =  A )
43abbii 2305 . . . . . 6  |-  { y  |  A  =  U. y }  =  {
y  |  U. y  =  A }
52, 4sseqtri 3204 . . . . 5  |-  { y  e.  Top  |  A  =  U. y }  C_  { y  |  U. y  =  A }
6 pwpwssunieq 3990 . . . . 5  |-  { y  |  U. y  =  A }  C_  ~P ~P A
75, 6sstri 3179 . . . 4  |-  { y  e.  Top  |  A  =  U. y }  C_  ~P ~P A
8 pwexg 4195 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
98pwexd 4196 . . . 4  |-  ( A  e.  V  ->  ~P ~P A  e.  _V )
10 ssexg 4157 . . . 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 414 . . 3  |-  ( A  e.  V  ->  { y  e.  Top  |  A  =  U. y }  e.  _V )
12 eqeq1 2196 . . . . 5  |-  ( x  =  A  ->  (
x  =  U. y  <->  A  =  U. y ) )
1312rabbidv 2741 . . . 4  |-  ( x  =  A  ->  { y  e.  Top  |  x  =  U. y }  =  { y  e. 
Top  |  A  =  U. y } )
14 df-topon 13915 . . . 4  |- TopOn  =  ( x  e.  _V  |->  { y  e.  Top  |  x  =  U. y } )
1513, 14fvmptg 5609 . . 3  |-  ( ( A  e.  _V  /\  { y  e.  Top  |  A  =  U. y }  e.  _V )  ->  (TopOn `  A )  =  { y  e.  Top  |  A  =  U. y } )
161, 11, 15syl2anc 411 . 2  |-  ( A  e.  V  ->  (TopOn `  A )  =  {
y  e.  Top  |  A  =  U. y } )
1716, 7eqsstrdi 3222 1  |-  ( A  e.  V  ->  (TopOn `  A )  C_  ~P ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160   {cab 2175   {crab 2472   _Vcvv 2752    C_ wss 3144   ~Pcpw 3590   U.cuni 3824   ` cfv 5232   Topctop 13901  TopOnctopon 13914
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5234  df-fv 5240  df-topon 13915
This theorem is referenced by: (None)
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