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Theorem toponsspwpwg 11774
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 2631 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
2 rabssab 3109 . . . . . 6  |-  { y  e.  Top  |  A  =  U. y }  C_  { y  |  A  = 
U. y }
3 eqcom 2091 . . . . . . 7  |-  ( A  =  U. y  <->  U. y  =  A )
43abbii 2204 . . . . . 6  |-  { y  |  A  =  U. y }  =  {
y  |  U. y  =  A }
52, 4sseqtri 3059 . . . . 5  |-  { y  e.  Top  |  A  =  U. y }  C_  { y  |  U. y  =  A }
6 pwpwssunieq 3823 . . . . 5  |-  { y  |  U. y  =  A }  C_  ~P ~P A
75, 6sstri 3035 . . . 4  |-  { y  e.  Top  |  A  =  U. y }  C_  ~P ~P A
8 pwexg 4021 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
98pwexd 4022 . . . 4  |-  ( A  e.  V  ->  ~P ~P A  e.  _V )
10 ssexg 3984 . . . 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 406 . . 3  |-  ( A  e.  V  ->  { y  e.  Top  |  A  =  U. y }  e.  _V )
12 eqeq1 2095 . . . . 5  |-  ( x  =  A  ->  (
x  =  U. y  <->  A  =  U. y ) )
1312rabbidv 2609 . . . 4  |-  ( x  =  A  ->  { y  e.  Top  |  x  =  U. y }  =  { y  e. 
Top  |  A  =  U. y } )
14 df-topon 11764 . . . 4  |- TopOn  =  ( x  e.  _V  |->  { y  e.  Top  |  x  =  U. y } )
1513, 14fvmptg 5393 . . 3  |-  ( ( A  e.  _V  /\  { y  e.  Top  |  A  =  U. y }  e.  _V )  ->  (TopOn `  A )  =  { y  e.  Top  |  A  =  U. y } )
161, 11, 15syl2anc 404 . 2  |-  ( A  e.  V  ->  (TopOn `  A )  =  {
y  e.  Top  |  A  =  U. y } )
1716, 7syl6eqss 3077 1  |-  ( A  e.  V  ->  (TopOn `  A )  C_  ~P ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290    e. wcel 1439   {cab 2075   {crab 2364   _Vcvv 2620    C_ wss 3000   ~Pcpw 3433   U.cuni 3659   ` cfv 5028   Topctop 11750  TopOnctopon 11763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-iota 4993  df-fun 5030  df-fv 5036  df-topon 11764
This theorem is referenced by: (None)
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