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Theorem lmfss 17360
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 17359 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  F  e.  ( X  ^pm  CC ) )
2 toponmax 16993 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
3 cnex 9071 . . . . 5  |-  CC  e.  _V
4 elpmg 7032 . . . . 5  |-  ( ( X  e.  J  /\  CC  e.  _V )  -> 
( F  e.  ( X  ^pm  CC )  <->  ( Fun  F  /\  F  C_  ( CC  X.  X
) ) ) )
52, 3, 4sylancl 644 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( F  e.  ( X  ^pm  CC ) 
<->  ( Fun  F  /\  F  C_  ( CC  X.  X ) ) ) )
65adantr 452 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  -> 
( F  e.  ( X  ^pm  CC )  <->  ( Fun  F  /\  F  C_  ( CC  X.  X
) ) ) )
71, 6mpbid 202 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  -> 
( Fun  F  /\  F  C_  ( CC  X.  X ) ) )
87simprd 450 1  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  F  C_  ( CC  X.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   _Vcvv 2956    C_ wss 3320   class class class wbr 4212    X. cxp 4876   Fun wfun 5448   ` cfv 5454  (class class class)co 6081    ^pm cpm 7019   CCcc 8988  TopOnctopon 16959   ~~> tclm 17290
This theorem is referenced by:  lmss  17362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-pm 7021  df-top 16963  df-topon 16966  df-lm 17293
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