MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  flimss1 Structured version   Unicode version

Theorem flimss1 17997
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 2435 . . . . . . 7  |-  U. K  =  U. K
21flimelbas 17992 . . . . . 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 17905 . . . . . . 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 17993 . . . . . . . 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 17905 . . . . . . 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 2469 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  X  =  U. K )
123, 11eleqtrrd 2512 . . . 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 16983 . . . . . . 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 17989 . . . . . . 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 16984 . . . . . . . 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 2469 . . . . . 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 2435 . . . . . . 7  |-  U. J  =  U. J
2322, 1topssnei 17180 . . . . . 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 17990 . . . . . 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 3350 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  ( ( nei `  J ) `  { x } ) 
C_  F )
28 elflim 17995 . . . . 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 3346 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 1652    e. wcel 1725    C_ wss 3312   {csn 3806   U.cuni 4007   ` cfv 5446  (class class class)co 6073   Topctop 16950  TopOnctopon 16951   neicnei 17153   Filcfil 17869    fLim cflim 17958
This theorem is referenced by:  flimcf  18006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-fbas 16691  df-top 16955  df-topon 16958  df-ntr 17076  df-nei 17154  df-fil 17870  df-flim 17963
  Copyright terms: Public domain W3C validator