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Theorem lmcls 17030
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 16989 . . . . 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 201 . . . 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 969 . . 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 11835 . . . . . 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 3509 . . . . . . . . . 10  |-  ( ( ( F `  k
)  e.  u  /\  ( F `  k )  e.  S )  -> 
( u  i^i  S
)  =/=  (/) )
1110a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k )  e.  u  /\  ( F `  k
)  e.  S )  ->  ( u  i^i 
S )  =/=  (/) ) )
129, 11mpan2d 655 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  e.  u  -> 
( u  i^i  S
)  =/=  (/) ) )
1312adantld 453 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  dom  F  /\  ( F `  k )  e.  u
)  ->  ( u  i^i  S )  =/=  (/) ) )
1413rexlimdva 2667 . . . . . 6  |-  ( ph  ->  ( E. k  e.  Z  ( k  e. 
dom  F  /\  ( F `  k )  e.  u )  ->  (
u  i^i  S )  =/=  (/) ) )
158, 14syl5 28 . . . . 5  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  u
)  ->  ( u  i^i  S )  =/=  (/) ) )
1615imim2d 48 . . . 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 2622 . . 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 14 . 2  |-  ( ph  ->  A. u  e.  J  ( P  e.  u  ->  ( u  i^i  S
)  =/=  (/) ) )
19 topontop 16664 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
202, 19syl 15 . . 3  |-  ( ph  ->  J  e.  Top )
21 lmcls.8 . . . 4  |-  ( ph  ->  S  C_  X )
22 toponuni 16665 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
232, 22syl 15 . . . 4  |-  ( ph  ->  X  =  U. J
)
2421, 23sseqtrd 3214 . . 3  |-  ( ph  ->  S  C_  U. J )
25 lmcl 17025 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  P  e.  X )
262, 1, 25syl2anc 642 . . . 4  |-  ( ph  ->  P  e.  X )
2726, 23eleqtrd 2359 . . 3  |-  ( ph  ->  P  e.  U. J
)
28 eqid 2283 . . . 4  |-  U. J  =  U. J
2928elcls 16810 . . 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 1182 . 2  |-  ( ph  ->  ( P  e.  ( ( cls `  J
) `  S )  <->  A. u  e.  J  ( P  e.  u  -> 
( u  i^i  S
)  =/=  (/) ) ) )
3118, 30mpbird 223 1  |-  ( ph  ->  P  e.  ( ( cls `  J ) `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455   U.cuni 3827   class class class wbr 4023   dom cdm 4689   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   CCcc 8735   ZZcz 10024   ZZ>=cuz 10230   Topctop 16631  TopOnctopon 16632   clsccl 16755   ~~> tclm 16956
This theorem is referenced by:  lmcld  17031  1stcelcls  17187  caublcls  18734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-neg 9040  df-z 10025  df-uz 10231  df-top 16636  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-lm 16959
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