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Theorem lmcls 17046
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 17005 . . . . 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 11851 . . . . . 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 3522 . . . . . . . . . 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 2680 . . . . . 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 2635 . . 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 16680 . . . 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 16681 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
232, 22syl 15 . . . 4  |-  ( ph  ->  X  =  U. J
)
2421, 23sseqtrd 3227 . . 3  |-  ( ph  ->  S  C_  U. J )
25 lmcl 17041 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  P  e.  X )
262, 1, 25syl2anc 642 . . . 4  |-  ( ph  ->  P  e.  X )
2726, 23eleqtrd 2372 . . 3  |-  ( ph  ->  P  e.  U. J
)
28 eqid 2296 . . . 4  |-  U. J  =  U. J
2928elcls 16826 . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   (/)c0 3468   U.cuni 3843   class class class wbr 4039   dom cdm 4705   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   CCcc 8751   ZZcz 10040   ZZ>=cuz 10246   Topctop 16647  TopOnctopon 16648   clsccl 16771   ~~> tclm 16972
This theorem is referenced by:  lmcld  17047  1stcelcls  17203  caublcls  18750
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
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 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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  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-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-neg 9056  df-z 10041  df-uz 10247  df-top 16652  df-topon 16655  df-cld 16772  df-ntr 16773  df-cls 16774  df-lm 16975
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