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Theorem flimcf 17693
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 734 . . . . . . 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 732 . . . . . . 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 731 . . . . . . 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 17684 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  f  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( K  fLim  f )  C_  ( J  fLim  f ) )
51, 2, 3, 4syl3anc 1182 . . . . . 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 733 . . . . . 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 3194 . . . . 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 598 . . . 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 3198 . . 3  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  /\  J  C_  K )  /\  f  e.  ( Fil `  X ) )  -> 
( K  fLim  f
)  C_  ( J  fLim  f ) )
109ralrimiva 2639 . 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 735 . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . 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 16683 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
1512, 13, 14syl2anc 642 . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . 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 3194 . . . . . . . . . . . . 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 3776 . . . . . . . . . . . 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 3756 . . . . . . . . . . . . 13  |-  ( y  e.  X  ->  { y }  =/=  (/) )
2017, 19syl 15 . . . . . . . . . . . 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 17591 . . . . . . . . . . . 12  |-  ( ( K  e.  (TopOn `  X )  /\  {
y }  C_  X  /\  { y }  =/=  (/) )  ->  ( ( nei `  K ) `  { y } )  e.  ( Fil `  X
) )
2211, 18, 20, 21syl3anc 1182 . . . . . . . . . . 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 731 . . . . . . . . . . 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 5882 . . . . . . . . . . . . 13  |-  ( f  =  ( ( nei `  K ) `  {
y } )  -> 
( K  fLim  f
)  =  ( K 
fLim  ( ( nei `  K ) `  {
y } ) ) )
25 oveq2 5882 . . . . . . . . . . . . 13  |-  ( f  =  ( ( nei `  K ) `  {
y } )  -> 
( J  fLim  f
)  =  ( J 
fLim  ( ( nei `  K ) `  {
y } ) ) )
2624, 25sseq12d 3220 . . . . . . . . . . . 12  |-  ( f  =  ( ( nei `  K ) `  {
y } )  -> 
( ( K  fLim  f )  C_  ( J  fLim  f )  <->  ( K  fLim  ( ( nei `  K
) `  { y } ) )  C_  ( J  fLim  ( ( nei `  K ) `
 { y } ) ) ) )
2726rspcv 2893 . . . . . . . . . . 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 56 . . . . . . . . . 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 17685 . . . . . . . . . . 11  |-  ( ( K  e.  (TopOn `  X )  /\  y  e.  X )  ->  y  e.  ( K  fLim  (
( nei `  K
) `  { y } ) ) )
3011, 17, 29syl2anc 642 . . . . . . . . . 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 3194 . . . . . . . . 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 17677 . . . . . . . . 9  |-  ( y  e.  ( J  fLim  ( ( nei `  K
) `  { y } ) )  -> 
( ( nei `  J
) `  { y } )  C_  (
( nei `  K
) `  { y } ) )
3331, 32syl 15 . . . . . . . 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 16680 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
3512, 34syl 15 . . . . . . . . 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 16872 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  x  e.  J  /\  y  e.  x )  ->  x  e.  ( ( nei `  J ) `
 { y } ) )
3735, 13, 16, 36syl3anc 1182 . . . . . . . 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 3194 . . . . . . 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 629 . . . . . 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 2639 . . . . 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 735 . . . . . 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 16680 . . . . . 6  |-  ( K  e.  (TopOn `  X
)  ->  K  e.  Top )
43 opnnei 16873 . . . . . 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 223 . . . 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 423 . . 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 3198 . 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 805 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   {csn 3653   ` cfv 5271  (class class class)co 5874   Topctop 16647  TopOnctopon 16648   neicnei 16850   Filcfil 17556    fLim cflim 17645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-top 16652  df-topon 16655  df-ntr 16773  df-nei 16851  df-fbas 17536  df-fil 17557  df-flim 17650
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