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Theorem lmfss 13829
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 13828 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  F  e.  ( X  ^pm  CC ) )
2 toponmax 13610 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
3 cnex 7937 . . . . 5  |-  CC  e.  _V
4 elpmg 6666 . . . . 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 2739    C_ wss 3131   class class class wbr 4005    X. cxp 4626   Fun wfun 5212   ` cfv 5218  (class class class)co 5877    ^pm cpm 6651   CCcc 7811  TopOnctopon 13595   ~~> tclm 13772
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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-pm 6653  df-top 13583  df-topon 13596  df-lm 13775
This theorem is referenced by:  lmss  13831
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