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Theorem flimcf 20775
Description: Fineness is properly characterized by the property that every limit point of a filter in the finer topology is a limit point in the coarser topology. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
flimcf  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  ->  ( J  C_  K  <->  A. f  e.  ( Fil `  X ) ( K  fLim  f
)  C_  ( J  fLim  f ) ) )
Distinct variable groups:    f, J    f, K    f, X

Proof of Theorem flimcf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplll 760 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  J  C_  K )  /\  ( f  e.  ( Fil `  X )  /\  x  e.  ( K  fLim  f )
) )  ->  J  e.  (TopOn `  X )
)
2 simprl 756 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  J  C_  K )  /\  ( f  e.  ( Fil `  X )  /\  x  e.  ( K  fLim  f )
) )  ->  f  e.  ( Fil `  X
) )
3 simplr 754 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  J  C_  K )  /\  ( f  e.  ( Fil `  X )  /\  x  e.  ( K  fLim  f )
) )  ->  J  C_  K )
4 flimss1 20766 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  f  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( K  fLim  f )  C_  ( J  fLim  f ) )
51, 2, 3, 4syl3anc 1230 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  J  C_  K )  /\  ( f  e.  ( Fil `  X )  /\  x  e.  ( K  fLim  f )
) )  ->  ( K  fLim  f )  C_  ( J  fLim  f ) )
6 simprr 758 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  J  C_  K )  /\  ( f  e.  ( Fil `  X )  /\  x  e.  ( K  fLim  f )
) )  ->  x  e.  ( K  fLim  f
) )
75, 6sseldd 3443 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  J  C_  K )  /\  ( f  e.  ( Fil `  X )  /\  x  e.  ( K  fLim  f )
) )  ->  x  e.  ( J  fLim  f
) )
87expr 613 . . . 4  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  J  C_  K )  /\  f  e.  ( Fil `  X ) )  -> 
( x  e.  ( K  fLim  f )  ->  x  e.  ( J 
fLim  f ) ) )
98ssrdv 3448 . . 3  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  J  C_  K )  /\  f  e.  ( Fil `  X ) )  -> 
( K  fLim  f
)  C_  ( J  fLim  f ) )
109ralrimiva 2818 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  J  C_  K
)  ->  A. f  e.  ( Fil `  X
) ( K  fLim  f )  C_  ( J  fLim  f ) )
11 simpllr 761 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  ->  K  e.  (TopOn `  X
) )
12 simplll 760 . . . . . . . . . . . . . . 15  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  ->  J  e.  (TopOn `  X
) )
13 simprl 756 . . . . . . . . . . . . . . 15  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  ->  x  e.  J )
14 toponss 19722 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
1512, 13, 14syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  ->  x  C_  X )
16 simprr 758 . . . . . . . . . . . . . 14  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  -> 
y  e.  x )
1715, 16sseldd 3443 . . . . . . . . . . . . 13  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  -> 
y  e.  X )
1817snssd 4117 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  ->  { y }  C_  X )
19 snnzg 4089 . . . . . . . . . . . . 13  |-  ( y  e.  X  ->  { y }  =/=  (/) )
2017, 19syl 17 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  ->  { y }  =/=  (/) )
21 neifil 20673 . . . . . . . . . . . 12  |-  ( ( K  e.  (TopOn `  X )  /\  {
y }  C_  X  /\  { y }  =/=  (/) )  ->  ( ( nei `  K ) `  { y } )  e.  ( Fil `  X
) )
2211, 18, 20, 21syl3anc 1230 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  -> 
( ( nei `  K
) `  { y } )  e.  ( Fil `  X ) )
23 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  ->  A. f  e.  ( Fil `  X ) ( K  fLim  f )  C_  ( J  fLim  f
) )
24 oveq2 6286 . . . . . . . . . . . . 13  |-  ( f  =  ( ( nei `  K ) `  {
y } )  -> 
( K  fLim  f
)  =  ( K 
fLim  ( ( nei `  K ) `  {
y } ) ) )
25 oveq2 6286 . . . . . . . . . . . . 13  |-  ( f  =  ( ( nei `  K ) `  {
y } )  -> 
( J  fLim  f
)  =  ( J 
fLim  ( ( nei `  K ) `  {
y } ) ) )
2624, 25sseq12d 3471 . . . . . . . . . . . 12  |-  ( f  =  ( ( nei `  K ) `  {
y } )  -> 
( ( K  fLim  f )  C_  ( J  fLim  f )  <->  ( K  fLim  ( ( nei `  K
) `  { y } ) )  C_  ( J  fLim  ( ( nei `  K ) `
 { y } ) ) ) )
2726rspcv 3156 . . . . . . . . . . 11  |-  ( ( ( nei `  K
) `  { y } )  e.  ( Fil `  X )  ->  ( A. f  e.  ( Fil `  X
) ( K  fLim  f )  C_  ( J  fLim  f )  ->  ( K  fLim  ( ( nei `  K ) `  {
y } ) ) 
C_  ( J  fLim  ( ( nei `  K
) `  { y } ) ) ) )
2822, 23, 27sylc 59 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  -> 
( K  fLim  (
( nei `  K
) `  { y } ) )  C_  ( J  fLim  ( ( nei `  K ) `
 { y } ) ) )
29 neiflim 20767 . . . . . . . . . . 11  |-  ( ( K  e.  (TopOn `  X )  /\  y  e.  X )  ->  y  e.  ( K  fLim  (
( nei `  K
) `  { y } ) ) )
3011, 17, 29syl2anc 659 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  -> 
y  e.  ( K 
fLim  ( ( nei `  K ) `  {
y } ) ) )
3128, 30sseldd 3443 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  -> 
y  e.  ( J 
fLim  ( ( nei `  K ) `  {
y } ) ) )
32 flimneiss 20759 . . . . . . . . 9  |-  ( y  e.  ( J  fLim  ( ( nei `  K
) `  { y } ) )  -> 
( ( nei `  J
) `  { y } )  C_  (
( nei `  K
) `  { y } ) )
3331, 32syl 17 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  -> 
( ( nei `  J
) `  { y } )  C_  (
( nei `  K
) `  { y } ) )
34 topontop 19719 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
3512, 34syl 17 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  ->  J  e.  Top )
36 opnneip 19913 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  x  e.  J  /\  y  e.  x )  ->  x  e.  ( ( nei `  J ) `
 { y } ) )
3735, 13, 16, 36syl3anc 1230 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  ->  x  e.  ( ( nei `  J ) `  { y } ) )
3833, 37sseldd 3443 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  ( x  e.  J  /\  y  e.  x ) )  ->  x  e.  ( ( nei `  K ) `  { y } ) )
3938anassrs 646 . . . . . 6  |-  ( ( ( ( ( J  e.  (TopOn `  X
)  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X
) ( K  fLim  f )  C_  ( J  fLim  f ) )  /\  x  e.  J )  /\  y  e.  x
)  ->  x  e.  ( ( nei `  K
) `  { y } ) )
4039ralrimiva 2818 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  x  e.  J )  ->  A. y  e.  x  x  e.  ( ( nei `  K
) `  { y } ) )
41 simpllr 761 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  x  e.  J )  ->  K  e.  (TopOn `  X )
)
42 topontop 19719 . . . . . 6  |-  ( K  e.  (TopOn `  X
)  ->  K  e.  Top )
43 opnnei 19914 . . . . . 6  |-  ( K  e.  Top  ->  (
x  e.  K  <->  A. y  e.  x  x  e.  ( ( nei `  K
) `  { y } ) ) )
4441, 42, 433syl 18 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  x  e.  J )  ->  (
x  e.  K  <->  A. y  e.  x  x  e.  ( ( nei `  K
) `  { y } ) ) )
4540, 44mpbird 232 . . . 4  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  A. f  e.  ( Fil `  X ) ( K 
fLim  f )  C_  ( J  fLim  f ) )  /\  x  e.  J )  ->  x  e.  K )
4645ex 432 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  A. f  e.  ( Fil `  X
) ( K  fLim  f )  C_  ( J  fLim  f ) )  -> 
( x  e.  J  ->  x  e.  K ) )
4746ssrdv 3448 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  A. f  e.  ( Fil `  X
) ( K  fLim  f )  C_  ( J  fLim  f ) )  ->  J  C_  K )
4810, 47impbida 833 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  ->  ( J  C_  K  <->  A. f  e.  ( Fil `  X ) ( K  fLim  f
)  C_  ( J  fLim  f ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754    C_ wss 3414   (/)c0 3738   {csn 3972   ` cfv 5569  (class class class)co 6278   Topctop 19686  TopOnctopon 19687   neicnei 19891   Filcfil 20638    fLim cflim 20727
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
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 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-fbas 18736  df-top 19691  df-topon 19694  df-ntr 19813  df-nei 19892  df-fil 20639  df-flim 20732
This theorem is referenced by: (None)
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