ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sslm Unicode version

Theorem sslm 12453
Description: A finer topology has fewer convergent sequences (but the sequences that do converge, converge to the same value). (Contributed by Mario Carneiro, 15-Sep-2015.)
Assertion
Ref Expression
sslm  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( ~~> t `  K )  C_  ( ~~> t `  J )
)

Proof of Theorem sslm
Dummy variables  u  f  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idd 21 . . . . 5  |-  ( J 
C_  K  ->  (
f  e.  ( X 
^pm  CC )  ->  f  e.  ( X  ^pm  CC ) ) )
2 idd 21 . . . . 5  |-  ( J 
C_  K  ->  (
x  e.  X  ->  x  e.  X )
)
3 ssralv 3165 . . . . 5  |-  ( J 
C_  K  ->  ( A. u  e.  K  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y
) : y --> u )  ->  A. u  e.  J  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y ) : y --> u ) ) )
41, 2, 33anim123d 1298 . . . 4  |-  ( J 
C_  K  ->  (
( f  e.  ( X  ^pm  CC )  /\  x  e.  X  /\  A. u  e.  K  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y
) : y --> u ) )  ->  (
f  e.  ( X 
^pm  CC )  /\  x  e.  X  /\  A. u  e.  J  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y ) : y --> u ) ) ) )
54ssopab2dv 4207 . . 3  |-  ( J 
C_  K  ->  { <. f ,  x >.  |  ( f  e.  ( X 
^pm  CC )  /\  x  e.  X  /\  A. u  e.  K  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y ) : y --> u ) ) } 
C_  { <. f ,  x >.  |  (
f  e.  ( X 
^pm  CC )  /\  x  e.  X  /\  A. u  e.  J  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y ) : y --> u ) ) } )
653ad2ant3 1005 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  { <. f ,  x >.  |  (
f  e.  ( X 
^pm  CC )  /\  x  e.  X  /\  A. u  e.  K  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y ) : y --> u ) ) } 
C_  { <. f ,  x >.  |  (
f  e.  ( X 
^pm  CC )  /\  x  e.  X  /\  A. u  e.  J  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y ) : y --> u ) ) } )
7 lmfval 12398 . . 3  |-  ( K  e.  (TopOn `  X
)  ->  ( ~~> t `  K )  =  { <. f ,  x >.  |  ( f  e.  ( X  ^pm  CC )  /\  x  e.  X  /\  A. u  e.  K  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y
) : y --> u ) ) } )
873ad2ant2 1004 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( ~~> t `  K )  =  { <. f ,  x >.  |  ( f  e.  ( X  ^pm  CC )  /\  x  e.  X  /\  A. u  e.  K  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y
) : y --> u ) ) } )
9 lmfval 12398 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( ~~> t `  J )  =  { <. f ,  x >.  |  ( f  e.  ( X  ^pm  CC )  /\  x  e.  X  /\  A. u  e.  J  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y
) : y --> u ) ) } )
1093ad2ant1 1003 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( ~~> t `  J )  =  { <. f ,  x >.  |  ( f  e.  ( X  ^pm  CC )  /\  x  e.  X  /\  A. u  e.  J  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y
) : y --> u ) ) } )
116, 8, 103sstr4d 3146 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( ~~> t `  K )  C_  ( ~~> t `  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 963    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418    C_ wss 3075   {copab 3995   ran crn 4547    |` cres 4548   -->wf 5126   ` cfv 5130  (class class class)co 5781    ^pm cpm 6550   CCcc 7641   ZZ>=cuz 9349  TopOnctopon 12214   ~~> tclm 12393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-cnex 7734
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-pm 6552  df-top 12202  df-topon 12215  df-lm 12396
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator