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Theorem lmfss 19967
Description: Inclusion of a function having a limit (used to ensure the limit relation is a set, under our definition). (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
Assertion
Ref Expression
lmfss  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  F  C_  ( CC  X.  X ) )

Proof of Theorem lmfss
StepHypRef Expression
1 lmfpm 19966 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  F  e.  ( X  ^pm  CC ) )
2 toponmax 19599 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
3 cnex 9562 . . . . 5  |-  CC  e.  _V
4 elpmg 7427 . . . . 5  |-  ( ( X  e.  J  /\  CC  e.  _V )  -> 
( F  e.  ( X  ^pm  CC )  <->  ( Fun  F  /\  F  C_  ( CC  X.  X
) ) ) )
52, 3, 4sylancl 660 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( F  e.  ( X  ^pm  CC ) 
<->  ( Fun  F  /\  F  C_  ( CC  X.  X ) ) ) )
65adantr 463 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  -> 
( F  e.  ( X  ^pm  CC )  <->  ( Fun  F  /\  F  C_  ( CC  X.  X
) ) ) )
71, 6mpbid 210 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  -> 
( Fun  F  /\  F  C_  ( CC  X.  X ) ) )
87simprd 461 1  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  F  C_  ( CC  X.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1823   _Vcvv 3106    C_ wss 3461   class class class wbr 4439    X. cxp 4986   Fun wfun 5564   ` cfv 5570  (class class class)co 6270    ^pm cpm 7413   CCcc 9479  TopOnctopon 19565   ~~> tclm 19897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-pm 7415  df-top 19569  df-topon 19572  df-lm 19900
This theorem is referenced by:  lmss  19969
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