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

Theorem flimss1 20764
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 2402 . . . . . . 7  |-  U. K  =  U. K
21flimelbas 20759 . . . . . 6  |-  ( x  e.  ( K  fLim  F )  ->  x  e.  U. K )
32adantl 464 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  x  e.  U. K )
4 simpl2 1001 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  F  e.  ( Fil `  X ) )
5 filunibas 20672 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
64, 5syl 17 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  U. F  =  X )
71flimfil 20760 . . . . . . . 8  |-  ( x  e.  ( K  fLim  F )  ->  F  e.  ( Fil `  U. K
) )
87adantl 464 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  F  e.  ( Fil `  U. K
) )
9 filunibas 20672 . . . . . . 7  |-  ( F  e.  ( Fil `  U. K )  ->  U. F  =  U. K )
108, 9syl 17 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  U. F  = 
U. K )
116, 10eqtr3d 2445 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  X  =  U. K )
123, 11eleqtrrd 2493 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  x  e.  X )
13 simpl1 1000 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  J  e.  (TopOn `  X ) )
14 topontop 19717 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
1513, 14syl 17 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  J  e.  Top )
16 flimtop 20756 . . . . . . 7  |-  ( x  e.  ( K  fLim  F )  ->  K  e.  Top )
1716adantl 464 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  K  e.  Top )
18 toponuni 19718 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1913, 18syl 17 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  X  =  U. J )
2019, 11eqtr3d 2445 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  U. J  = 
U. K )
21 simpl3 1002 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  J  C_  K
)
22 eqid 2402 . . . . . . 7  |-  U. J  =  U. J
2322, 1topssnei 19916 . . . . . 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 1233 . . . . 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 20757 . . . . . 6  |-  ( x  e.  ( K  fLim  F )  ->  ( ( nei `  K ) `  { x } ) 
C_  F )
2625adantl 464 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  ( ( nei `  K ) `  { x } ) 
C_  F )
2724, 26sstrd 3451 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  ( ( nei `  J ) `  { x } ) 
C_  F )
28 elflim 20762 . . . . 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 659 . . . 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 923 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  x  e.  ( J  fLim  F ) )
3130ex 432 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( x  e.  ( K  fLim  F
)  ->  x  e.  ( J  fLim  F ) ) )
3231ssrdv 3447 1  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( K  fLim  F )  C_  ( J  fLim  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    C_ wss 3413   {csn 3971   U.cuni 4190   ` cfv 5568  (class class class)co 6277   Topctop 19684  TopOnctopon 19685   neicnei 19889   Filcfil 20636    fLim cflim 20725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-fbas 18734  df-top 19689  df-topon 19692  df-ntr 19811  df-nei 19890  df-fil 20637  df-flim 20730
This theorem is referenced by:  flimcf  20773
  Copyright terms: Public domain W3C validator