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Theorem lmfss 13637
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 13636 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  F  e.  ( X  ^pm  CC ) )
2 toponmax 13416 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
3 cnex 7934 . . . . 5  |-  CC  e.  _V
4 elpmg 6663 . . . . 5  |-  ( ( X  e.  J  /\  CC  e.  _V )  -> 
( F  e.  ( X  ^pm  CC )  <->  ( Fun  F  /\  F  C_  ( CC  X.  X
) ) ) )
52, 3, 4sylancl 413 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( F  e.  ( X  ^pm  CC ) 
<->  ( Fun  F  /\  F  C_  ( CC  X.  X ) ) ) )
65adantr 276 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  -> 
( F  e.  ( X  ^pm  CC )  <->  ( Fun  F  /\  F  C_  ( CC  X.  X
) ) ) )
71, 6mpbid 147 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  -> 
( Fun  F  /\  F  C_  ( CC  X.  X ) ) )
87simprd 114 1  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  F  C_  ( CC  X.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2148   _Vcvv 2737    C_ wss 3129   class class class wbr 4003    X. cxp 4624   Fun wfun 5210   ` cfv 5216  (class class class)co 5874    ^pm cpm 6648   CCcc 7808  TopOnctopon 13401   ~~> tclm 13580
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-in1 614  ax-in2 615  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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  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-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  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 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-pm 6650  df-top 13389  df-topon 13402  df-lm 13583
This theorem is referenced by:  lmss  13639
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