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Theorem lmfpm 17321
Description: If  F converges, then  F is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.)
Assertion
Ref Expression
lmfpm  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  F  e.  ( X  ^pm  CC ) )

Proof of Theorem lmfpm
Dummy variables  y  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 20 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  (TopOn `  X ) )
21lmbr 17284 . . 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 471 . 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 ) ) )
43simp1d 969 1  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  F  e.  ( X  ^pm  CC ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1721   A.wral 2674   E.wrex 2675   class class class wbr 4180   ran crn 4846    |` cres 4847   -->wf 5417   ` cfv 5421  (class class class)co 6048    ^pm cpm 6986   CCcc 8952   ZZ>=cuz 10452  TopOnctopon 16922   ~~> tclm 17252
This theorem is referenced by:  lmfss  17322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-top 16926  df-topon 16929  df-lm 17255
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