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Theorem lmcl 13616
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 13584 . . 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 296 . 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 1010 1  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  P  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    e. wcel 2148   A.wral 2455   E.wrex 2456   class class class wbr 4002   ran crn 4626    |` cres 4627   -->wf 5211   ` cfv 5215  (class class class)co 5872    ^pm cpm 6646   CCcc 7806   ZZ>=cuz 9524  TopOnctopon 13379   ~~> tclm 13558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-cnex 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-pm 6648  df-top 13367  df-topon 13380  df-lm 13561
This theorem is referenced by:  lmss  13617  lmff  13620  lmcn  13622
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