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Theorem flimcf 20211
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 757 . . . . . . 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 755 . . . . . . 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 20202 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  f  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( K  fLim  f )  C_  ( J  fLim  f ) )
51, 2, 3, 4syl3anc 1223 . . . . . 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 756 . . . . . 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 3498 . . . . 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 615 . . . 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 3503 . . 3  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  J  C_  K )  /\  f  e.  ( Fil `  X ) )  -> 
( K  fLim  f
)  C_  ( J  fLim  f ) )
109ralrimiva 2871 . 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 758 . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . 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 19190 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
1512, 13, 14syl2anc 661 . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . 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 3498 . . . . . . . . . . . . 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 4165 . . . . . . . . . . . 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 4137 . . . . . . . . . . . . 13  |-  ( y  e.  X  ->  { y }  =/=  (/) )
2017, 19syl 16 . . . . . . . . . . . 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 20109 . . . . . . . . . . . 12  |-  ( ( K  e.  (TopOn `  X )  /\  {
y }  C_  X  /\  { y }  =/=  (/) )  ->  ( ( nei `  K ) `  { y } )  e.  ( Fil `  X
) )
2211, 18, 20, 21syl3anc 1223 . . . . . . . . . . 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 6283 . . . . . . . . . . . . 13  |-  ( f  =  ( ( nei `  K ) `  {
y } )  -> 
( K  fLim  f
)  =  ( K 
fLim  ( ( nei `  K ) `  {
y } ) ) )
25 oveq2 6283 . . . . . . . . . . . . 13  |-  ( f  =  ( ( nei `  K ) `  {
y } )  -> 
( J  fLim  f
)  =  ( J 
fLim  ( ( nei `  K ) `  {
y } ) ) )
2624, 25sseq12d 3526 . . . . . . . . . . . 12  |-  ( f  =  ( ( nei `  K ) `  {
y } )  -> 
( ( K  fLim  f )  C_  ( J  fLim  f )  <->  ( K  fLim  ( ( nei `  K
) `  { y } ) )  C_  ( J  fLim  ( ( nei `  K ) `
 { y } ) ) ) )
2726rspcv 3203 . . . . . . . . . . 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 60 . . . . . . . . . 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 20203 . . . . . . . . . . 11  |-  ( ( K  e.  (TopOn `  X )  /\  y  e.  X )  ->  y  e.  ( K  fLim  (
( nei `  K
) `  { y } ) ) )
3011, 17, 29syl2anc 661 . . . . . . . . . 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 3498 . . . . . . . . 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 20195 . . . . . . . . 9  |-  ( y  e.  ( J  fLim  ( ( nei `  K
) `  { y } ) )  -> 
( ( nei `  J
) `  { y } )  C_  (
( nei `  K
) `  { y } ) )
3331, 32syl 16 . . . . . . . 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 19187 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
3512, 34syl 16 . . . . . . . . 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 19379 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  x  e.  J  /\  y  e.  x )  ->  x  e.  ( ( nei `  J ) `
 { y } ) )
3735, 13, 16, 36syl3anc 1223 . . . . . . . 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 3498 . . . . . . 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 648 . . . . . 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 2871 . . . . 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 758 . . . . . 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 19187 . . . . . 6  |-  ( K  e.  (TopOn `  X
)  ->  K  e.  Top )
43 opnnei 19380 . . . . . 6  |-  ( K  e.  Top  ->  (
x  e.  K  <->  A. y  e.  x  x  e.  ( ( nei `  K
) `  { y } ) ) )
4441, 42, 433syl 20 . . . . 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 434 . . 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 3503 . 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 829 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 369    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807    C_ wss 3469   (/)c0 3778   {csn 4020   ` cfv 5579  (class class class)co 6275   Topctop 19154  TopOnctopon 19155   neicnei 19357   Filcfil 20074    fLim cflim 20163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-fbas 18180  df-top 19159  df-topon 19162  df-ntr 19280  df-nei 19358  df-fil 20075  df-flim 20168
This theorem is referenced by: (None)
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