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

Theorem sslm 17355
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 22 . . . . 5  |-  ( J 
C_  K  ->  (
f  e.  ( X 
^pm  CC )  ->  f  e.  ( X  ^pm  CC ) ) )
2 idd 22 . . . . 5  |-  ( J 
C_  K  ->  (
x  e.  X  ->  x  e.  X )
)
3 ssralv 3399 . . . . 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 1261 . . . 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 4475 . . 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 980 . 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 17288 . . 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 979 . 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 17288 . . 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 978 . 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 3383 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 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   {copab 4257   ran crn 4871    |` cres 4872   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^pm cpm 7011   CCcc 8980   ZZ>=cuz 10480  TopOnctopon 16951   ~~> tclm 17282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-top 16955  df-topon 16958  df-lm 17285
  Copyright terms: Public domain W3C validator