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Theorem toponsspwpwg 12100
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 2671 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
2 rabssab 3154 . . . . . 6  |-  { y  e.  Top  |  A  =  U. y }  C_  { y  |  A  = 
U. y }
3 eqcom 2119 . . . . . . 7  |-  ( A  =  U. y  <->  U. y  =  A )
43abbii 2233 . . . . . 6  |-  { y  |  A  =  U. y }  =  {
y  |  U. y  =  A }
52, 4sseqtri 3101 . . . . 5  |-  { y  e.  Top  |  A  =  U. y }  C_  { y  |  U. y  =  A }
6 pwpwssunieq 3871 . . . . 5  |-  { y  |  U. y  =  A }  C_  ~P ~P A
75, 6sstri 3076 . . . 4  |-  { y  e.  Top  |  A  =  U. y }  C_  ~P ~P A
8 pwexg 4074 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
98pwexd 4075 . . . 4  |-  ( A  e.  V  ->  ~P ~P A  e.  _V )
10 ssexg 4037 . . . 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 2124 . . . . 5  |-  ( x  =  A  ->  (
x  =  U. y  <->  A  =  U. y ) )
1312rabbidv 2649 . . . 4  |-  ( x  =  A  ->  { y  e.  Top  |  x  =  U. y }  =  { y  e. 
Top  |  A  =  U. y } )
14 df-topon 12089 . . . 4  |- TopOn  =  ( x  e.  _V  |->  { y  e.  Top  |  x  =  U. y } )
1513, 14fvmptg 5465 . . 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 3119 1  |-  ( A  e.  V  ->  (TopOn `  A )  C_  ~P ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   {cab 2103   {crab 2397   _Vcvv 2660    C_ wss 3041   ~Pcpw 3480   U.cuni 3706   ` cfv 5093   Topctop 12075  TopOnctopon 12088
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-topon 12089
This theorem is referenced by: (None)
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