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Theorem flimss1 17926
Description: A limit point of a filter is a limit point in a coarser topology. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
flimss1  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( K  fLim  F )  C_  ( J  fLim  F ) )

Proof of Theorem flimss1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2387 . . . . . . 7  |-  U. K  =  U. K
21flimelbas 17921 . . . . . 6  |-  ( x  e.  ( K  fLim  F )  ->  x  e.  U. K )
32adantl 453 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  x  e.  U. K )
4 simpl2 961 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  F  e.  ( Fil `  X ) )
5 filunibas 17834 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
64, 5syl 16 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  U. F  =  X )
71flimfil 17922 . . . . . . . 8  |-  ( x  e.  ( K  fLim  F )  ->  F  e.  ( Fil `  U. K
) )
87adantl 453 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  F  e.  ( Fil `  U. K
) )
9 filunibas 17834 . . . . . . 7  |-  ( F  e.  ( Fil `  U. K )  ->  U. F  =  U. K )
108, 9syl 16 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  U. F  = 
U. K )
116, 10eqtr3d 2421 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  X  =  U. K )
123, 11eleqtrrd 2464 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  x  e.  X )
13 simpl1 960 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  J  e.  (TopOn `  X ) )
14 topontop 16914 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
1513, 14syl 16 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  J  e.  Top )
16 flimtop 17918 . . . . . . 7  |-  ( x  e.  ( K  fLim  F )  ->  K  e.  Top )
1716adantl 453 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  K  e.  Top )
18 toponuni 16915 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1913, 18syl 16 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  X  =  U. J )
2019, 11eqtr3d 2421 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  U. J  = 
U. K )
21 simpl3 962 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  J  C_  K
)
22 eqid 2387 . . . . . . 7  |-  U. J  =  U. J
2322, 1topssnei 17111 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  U. J  =  U. K
)  /\  J  C_  K
)  ->  ( ( nei `  J ) `  { x } ) 
C_  ( ( nei `  K ) `  {
x } ) )
2415, 17, 20, 21, 23syl31anc 1187 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  ( ( nei `  J ) `  { x } ) 
C_  ( ( nei `  K ) `  {
x } ) )
25 flimneiss 17919 . . . . . 6  |-  ( x  e.  ( K  fLim  F )  ->  ( ( nei `  K ) `  { x } ) 
C_  F )
2625adantl 453 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  ( ( nei `  K ) `  { x } ) 
C_  F )
2724, 26sstrd 3301 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  ( ( nei `  J ) `  { x } ) 
C_  F )
28 elflim 17924 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  ( J 
fLim  F )  <->  ( x  e.  X  /\  (
( nei `  J
) `  { x } )  C_  F
) ) )
2913, 4, 28syl2anc 643 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  ( x  e.  ( J  fLim  F
)  <->  ( x  e.  X  /\  ( ( nei `  J ) `
 { x }
)  C_  F )
) )
3012, 27, 29mpbir2and 889 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  x  e.  ( J  fLim  F ) )
3130ex 424 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( x  e.  ( K  fLim  F
)  ->  x  e.  ( J  fLim  F ) ) )
3231ssrdv 3297 1  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( K  fLim  F )  C_  ( J  fLim  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3263   {csn 3757   U.cuni 3957   ` cfv 5394  (class class class)co 6020   Topctop 16881  TopOnctopon 16882   neicnei 17084   Filcfil 17798    fLim cflim 17887
This theorem is referenced by:  flimcf  17935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-fbas 16623  df-top 16886  df-topon 16889  df-ntr 17007  df-nei 17085  df-fil 17799  df-flim 17892
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