Theorem lmfss 15345

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

Step	Hyp	Ref	Expression
1		lmfpm 15344	. . 3 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> F e. ( X ^pm CC ) )
2		toponmax 15126	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
3		cnex 8303	. . . . 5 |- CC e. _V
4		elpmg 6938	. . . . 5 |- ( ( X e. J /\ CC e. _V ) -> ( F e. ( X ^pm CC ) <-> ( Fun F /\ F C_ ( CC X. X ) ) ) )
5	2, 3, 4	sylancl 417	. . . 4 |- ( J e. (TopOn ` X ) -> ( F e. ( X ^pm CC ) <-> ( Fun F /\ F C_ ( CC X. X ) ) ) )
6	5	adantr 276	. . 3 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> ( F e. ( X ^pm CC ) <-> ( Fun F /\ F C_ ( CC X. X ) ) ) )
7	1, 6	mpbid 147	. 2 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> ( Fun F /\ F C_ ( CC X. X ) ) )
8	7	simprd 114	1 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> F C_ ( CC X. X ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   _Vcvv 2821   C_ wss 3220   class class class wbr 4130   X. cxp 4772   Fun wfun 5371   ` cfv 5377  (class class class)co 6085   ^pm cpm 6923   CCcc 8177  TopOnctopon 15111   ~~> tclm 15288

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pm 6925  df-top 15099  df-topon 15112  df-lm 15291

This theorem is used by:  lmss  15347