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Theorem lmcl 17041
Description: Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
Assertion
Ref Expression
lmcl  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  P  e.  X )

Proof of Theorem lmcl
Dummy variables  y  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  (TopOn `  X ) )
21lmbr 17004 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( F
( ~~> t `  J
) P  <->  ( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. u  e.  J  ( P  e.  u  ->  E. y  e.  ran  ZZ>= ( F  |`  y ) : y --> u ) ) ) )
32biimpa 470 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  -> 
( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. u  e.  J  ( P  e.  u  ->  E. y  e.  ran  ZZ>= ( F  |`  y ) : y --> u ) ) )
43simp2d 968 1  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  P  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1696   A.wral 2556   E.wrex 2557   class class class wbr 4039   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 is referenced by:  lmss  17042  lmff  17045  lmcls  17046  lmcn  17049  lmmo  17124  1stccn  17205  1stckgenlem  17264  1stckgen  17265  cmetcaulem  18730  iscmet3lem2  18734  nvlmcl  21280  minvecolem4b  21473  minvecolem4  21475  axhcompl-zf  21594  heiborlem9  26646  bfplem2  26650  climreeq  27842
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-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-top 16652  df-topon 16655  df-lm 16975
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