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

Theorem flimss1 20209
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 2467 . . . . . . 7  |-  U. K  =  U. K
21flimelbas 20204 . . . . . 6  |-  ( x  e.  ( K  fLim  F )  ->  x  e.  U. K )
32adantl 466 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  x  e.  U. K )
4 simpl2 1000 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  F  e.  ( Fil `  X ) )
5 filunibas 20117 . . . . . . 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 20205 . . . . . . . 8  |-  ( x  e.  ( K  fLim  F )  ->  F  e.  ( Fil `  U. K
) )
87adantl 466 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  F  e.  ( Fil `  U. K
) )
9 filunibas 20117 . . . . . . 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 2510 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  X  =  U. K )
123, 11eleqtrrd 2558 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  x  e.  X )
13 simpl1 999 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  J  e.  (TopOn `  X ) )
14 topontop 19194 . . . . . . 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 20201 . . . . . . 7  |-  ( x  e.  ( K  fLim  F )  ->  K  e.  Top )
1716adantl 466 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  K  e.  Top )
18 toponuni 19195 . . . . . . . 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 2510 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  U. J  = 
U. K )
21 simpl3 1001 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  J  C_  K
)
22 eqid 2467 . . . . . . 7  |-  U. J  =  U. J
2322, 1topssnei 19391 . . . . . 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 1231 . . . . 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 20202 . . . . . 6  |-  ( x  e.  ( K  fLim  F )  ->  ( ( nei `  K ) `  { x } ) 
C_  F )
2625adantl 466 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  ( ( nei `  K ) `  { x } ) 
C_  F )
2724, 26sstrd 3514 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  ( ( nei `  J ) `  { x } ) 
C_  F )
28 elflim 20207 . . . . 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 661 . . . 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 920 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fLim  F ) )  ->  x  e.  ( J  fLim  F ) )
3130ex 434 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( x  e.  ( K  fLim  F
)  ->  x  e.  ( J  fLim  F ) ) )
3231ssrdv 3510 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   {csn 4027   U.cuni 4245   ` cfv 5586  (class class class)co 6282   Topctop 19161  TopOnctopon 19162   neicnei 19364   Filcfil 20081    fLim cflim 20170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-fbas 18187  df-top 19166  df-topon 19169  df-ntr 19287  df-nei 19365  df-fil 20082  df-flim 20175
This theorem is referenced by:  flimcf  20218
  Copyright terms: Public domain W3C validator