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Theorem lmcls 17366
Description: Any convergent sequence of points in a subset of a topological space converges to a point in the closure of the subset. (Contributed by Mario Carneiro, 30-Dec-2013.) (Revised by Mario Carneiro, 1-May-2014.)
Hypotheses
Ref Expression
lmff.1  |-  Z  =  ( ZZ>= `  M )
lmff.3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
lmff.4  |-  ( ph  ->  M  e.  ZZ )
lmcls.5  |-  ( ph  ->  F ( ~~> t `  J ) P )
lmcls.7  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  S )
lmcls.8  |-  ( ph  ->  S  C_  X )
Assertion
Ref Expression
lmcls  |-  ( ph  ->  P  e.  ( ( cls `  J ) `
 S ) )
Distinct variable groups:    k, F    k, J    k, M    P, k    S, k    ph, k    k, X    k, Z

Proof of Theorem lmcls
Dummy variables  j  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmcls.5 . . . . 5  |-  ( ph  ->  F ( ~~> t `  J ) P )
2 lmff.3 . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
3 lmff.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
4 lmff.4 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
52, 3, 4lmbr2 17323 . . . . 5  |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  ( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. u  e.  J  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  u ) ) ) ) )
61, 5mpbid 202 . . . 4  |-  ( ph  ->  ( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. u  e.  J  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  u ) ) ) )
76simp3d 971 . . 3  |-  ( ph  ->  A. u  e.  J  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  u ) ) )
83r19.2uz 12155 . . . . . 6  |-  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  u )  ->  E. k  e.  Z  ( k  e.  dom  F  /\  ( F `  k )  e.  u
) )
9 lmcls.7 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  S )
10 inelcm 3682 . . . . . . . . . 10  |-  ( ( ( F `  k
)  e.  u  /\  ( F `  k )  e.  S )  -> 
( u  i^i  S
)  =/=  (/) )
1110a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k )  e.  u  /\  ( F `  k
)  e.  S )  ->  ( u  i^i 
S )  =/=  (/) ) )
129, 11mpan2d 656 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  e.  u  -> 
( u  i^i  S
)  =/=  (/) ) )
1312adantld 454 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  dom  F  /\  ( F `  k )  e.  u
)  ->  ( u  i^i  S )  =/=  (/) ) )
1413rexlimdva 2830 . . . . . 6  |-  ( ph  ->  ( E. k  e.  Z  ( k  e. 
dom  F  /\  ( F `  k )  e.  u )  ->  (
u  i^i  S )  =/=  (/) ) )
158, 14syl5 30 . . . . 5  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  u
)  ->  ( u  i^i  S )  =/=  (/) ) )
1615imim2d 50 . . . 4  |-  ( ph  ->  ( ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  u
) )  ->  ( P  e.  u  ->  ( u  i^i  S )  =/=  (/) ) ) )
1716ralimdv 2785 . . 3  |-  ( ph  ->  ( A. u  e.  J  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  u
) )  ->  A. u  e.  J  ( P  e.  u  ->  ( u  i^i  S )  =/=  (/) ) ) )
187, 17mpd 15 . 2  |-  ( ph  ->  A. u  e.  J  ( P  e.  u  ->  ( u  i^i  S
)  =/=  (/) ) )
19 topontop 16991 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
202, 19syl 16 . . 3  |-  ( ph  ->  J  e.  Top )
21 lmcls.8 . . . 4  |-  ( ph  ->  S  C_  X )
22 toponuni 16992 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
232, 22syl 16 . . . 4  |-  ( ph  ->  X  =  U. J
)
2421, 23sseqtrd 3384 . . 3  |-  ( ph  ->  S  C_  U. J )
25 lmcl 17361 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  P  e.  X )
262, 1, 25syl2anc 643 . . . 4  |-  ( ph  ->  P  e.  X )
2726, 23eleqtrd 2512 . . 3  |-  ( ph  ->  P  e.  U. J
)
28 eqid 2436 . . . 4  |-  U. J  =  U. J
2928elcls 17137 . . 3  |-  ( ( J  e.  Top  /\  S  C_  U. J  /\  P  e.  U. J )  ->  ( P  e.  ( ( cls `  J
) `  S )  <->  A. u  e.  J  ( P  e.  u  -> 
( u  i^i  S
)  =/=  (/) ) ) )
3020, 24, 27, 29syl3anc 1184 . 2  |-  ( ph  ->  ( P  e.  ( ( cls `  J
) `  S )  <->  A. u  e.  J  ( P  e.  u  -> 
( u  i^i  S
)  =/=  (/) ) ) )
3118, 30mpbird 224 1  |-  ( ph  ->  P  e.  ( ( cls `  J ) `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706    i^i cin 3319    C_ wss 3320   (/)c0 3628   U.cuni 4015   class class class wbr 4212   dom cdm 4878   ` cfv 5454  (class class class)co 6081    ^pm cpm 7019   CCcc 8988   ZZcz 10282   ZZ>=cuz 10488   Topctop 16958  TopOnctopon 16959   clsccl 17082   ~~> tclm 17290
This theorem is referenced by:  lmcld  17367  1stcelcls  17524  caublcls  19261
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-pre-lttri 9064  ax-pre-lttrn 9065
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-neg 9294  df-z 10283  df-uz 10489  df-top 16963  df-topon 16966  df-cld 17083  df-ntr 17084  df-cls 17085  df-lm 17293
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