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Theorem flimss1 17668
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 2283 . . . . . . 7  |-  U. K  =  U. K
21flimelbas 17663 . . . . . 6  |-  ( x  e.  ( K  fLim  F )  ->  x  e.  U. K )
32adantl 452 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  x  e.  U. K )
4 simpl2 959 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  F  e.  ( Fil `  X ) )
5 filunibas 17576 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
64, 5syl 15 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  U. F  =  X )
71flimfil 17664 . . . . . . . 8  |-  ( x  e.  ( K  fLim  F )  ->  F  e.  ( Fil `  U. K
) )
87adantl 452 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  F  e.  ( Fil `  U. K
) )
9 filunibas 17576 . . . . . . 7  |-  ( F  e.  ( Fil `  U. K )  ->  U. F  =  U. K )
108, 9syl 15 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  U. F  = 
U. K )
116, 10eqtr3d 2317 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  X  =  U. K )
123, 11eleqtrrd 2360 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  x  e.  X )
13 simpl1 958 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  J  e.  (TopOn `  X ) )
14 topontop 16664 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
1513, 14syl 15 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  J  e.  Top )
16 flimtop 17660 . . . . . . 7  |-  ( x  e.  ( K  fLim  F )  ->  K  e.  Top )
1716adantl 452 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  K  e.  Top )
18 toponuni 16665 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1913, 18syl 15 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  X  =  U. J )
2019, 11eqtr3d 2317 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  U. J  = 
U. K )
21 simpl3 960 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  J  C_  K
)
22 eqid 2283 . . . . . . 7  |-  U. J  =  U. J
2322, 1topssnei 16861 . . . . . 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 1185 . . . . 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 17661 . . . . . 6  |-  ( x  e.  ( K  fLim  F )  ->  ( ( nei `  K ) `  { x } ) 
C_  F )
2625adantl 452 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  ( ( nei `  K ) `  { x } ) 
C_  F )
2724, 26sstrd 3189 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  ( ( nei `  J ) `  { x } ) 
C_  F )
28 elflim 17666 . . . . 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 642 . . . 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 888 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  x  e.  ( J  fLim  F ) )
3130ex 423 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( x  e.  ( K  fLim  F
)  ->  x  e.  ( J  fLim  F ) ) )
3231ssrdv 3185 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   {csn 3640   U.cuni 3827   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632   neicnei 16834   Filcfil 17540    fLim cflim 17629
This theorem is referenced by:  flimcf  17677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-top 16636  df-topon 16639  df-ntr 16757  df-nei 16835  df-fbas 17520  df-fil 17541  df-flim 17634
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