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Theorem sslm 17043
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 3250 . . . . 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 1259 . . . 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 4309 . . 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 978 . 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 16978 . . 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 977 . 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 16978 . . 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 976 . 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 3234 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 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   {copab 4092   ran crn 4706    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   CCcc 8751   ZZ>=cuz 10246  TopOnctopon 16648   ~~> tclm 16972
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-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-ral 2561  df-rex 2562  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-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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-top 16652  df-topon 16655  df-lm 16975
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