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Theorem lmcls 19031
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 18988 . . . . 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 210 . . . 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 1002 . . 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 12950 . . . . . 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 3834 . . . . . . . . . 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 674 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  e.  u  -> 
( u  i^i  S
)  =/=  (/) ) )
1312adantld 467 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  dom  F  /\  ( F `  k )  e.  u
)  ->  ( u  i^i  S )  =/=  (/) ) )
1413rexlimdva 2940 . . . . . 6  |-  ( ph  ->  ( E. k  e.  Z  ( k  e. 
dom  F  /\  ( F `  k )  e.  u )  ->  (
u  i^i  S )  =/=  (/) ) )
158, 14syl5 32 . . . . 5  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  u
)  ->  ( u  i^i  S )  =/=  (/) ) )
1615imim2d 52 . . . 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 2829 . . 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 18656 . . . 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 18657 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
232, 22syl 16 . . . 4  |-  ( ph  ->  X  =  U. J
)
2421, 23sseqtrd 3493 . . 3  |-  ( ph  ->  S  C_  U. J )
25 lmcl 19026 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  P  e.  X )
262, 1, 25syl2anc 661 . . . 4  |-  ( ph  ->  P  e.  X )
2726, 23eleqtrd 2541 . . 3  |-  ( ph  ->  P  e.  U. J
)
28 eqid 2451 . . . 4  |-  U. J  =  U. J
2928elcls 18802 . . 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 1219 . 2  |-  ( ph  ->  ( P  e.  ( ( cls `  J
) `  S )  <->  A. u  e.  J  ( P  e.  u  -> 
( u  i^i  S
)  =/=  (/) ) ) )
3118, 30mpbird 232 1  |-  ( ph  ->  P  e.  ( ( cls `  J ) `
 S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796    i^i cin 3428    C_ wss 3429   (/)c0 3738   U.cuni 4192   class class class wbr 4393   dom cdm 4941   ` cfv 5519  (class class class)co 6193    ^pm cpm 7318   CCcc 9384   ZZcz 10750   ZZ>=cuz 10965   Topctop 18623  TopOnctopon 18624   clsccl 18747   ~~> tclm 18955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-pre-lttri 9460  ax-pre-lttrn 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-er 7204  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-neg 9702  df-z 10751  df-uz 10966  df-top 18628  df-topon 18631  df-cld 18748  df-ntr 18749  df-cls 18750  df-lm 18958
This theorem is referenced by:  lmcld  19032  1stcelcls  19190  caublcls  20944
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