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Theorem lmcl 19924
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 22 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  (TopOn `  X ) )
21lmbr 19885 . . 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 484 . 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 1009 1  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  P  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1819   A.wral 2807   E.wrex 2808   class class class wbr 4456   ran crn 5009    |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^pm cpm 7439   CCcc 9507   ZZ>=cuz 11106  TopOnctopon 19521   ~~> tclm 19853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-top 19525  df-topon 19528  df-lm 19856
This theorem is referenced by:  lmss  19925  lmff  19928  lmcls  19929  lmcn  19932  lmmo  20007  1stccn  20089  1stckgenlem  20179  1stckgen  20180  cmetcaulem  21852  iscmet3lem2  21856  nvlmcl  25727  minvecolem4b  25920  minvecolem4  25922  axhcompl-zf  26041  heiborlem9  30477  bfplem2  30481  climreeq  31780
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