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Theorem lmfss 19027
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 19026 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  F  e.  ( X  ^pm  CC ) )
2 toponmax 18660 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
3 cnex 9469 . . . . 5  |-  CC  e.  _V
4 elpmg 7333 . . . . 5  |-  ( ( X  e.  J  /\  CC  e.  _V )  -> 
( F  e.  ( X  ^pm  CC )  <->  ( Fun  F  /\  F  C_  ( CC  X.  X
) ) ) )
52, 3, 4sylancl 662 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( F  e.  ( X  ^pm  CC ) 
<->  ( Fun  F  /\  F  C_  ( CC  X.  X ) ) ) )
65adantr 465 . . 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 463 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 369    e. wcel 1758   _Vcvv 3072    C_ wss 3431   class class class wbr 4395    X. cxp 4941   Fun wfun 5515   ` cfv 5521  (class class class)co 6195    ^pm cpm 7320   CCcc 9386  TopOnctopon 18626   ~~> tclm 18957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-pm 7322  df-top 18630  df-topon 18633  df-lm 18960
This theorem is referenced by:  lmss  19029
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