Theorem lmfss 15038

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 15037	. . 3 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> F e. ( X ^pm CC ) )
2		toponmax 14819	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
3		cnex 8199	. . . . 5 |- CC e. _V
4		elpmg 6876	. . . . 5 |- ( ( X e. J /\ CC e. _V ) -> ( F e. ( X ^pm CC ) <-> ( Fun F /\ F C_ ( CC X. X ) ) ) )
5	2, 3, 4	sylancl 413	. . . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2202   _Vcvv 2803   C_ wss 3201   class class class wbr 4093   X. cxp 4729   Fun wfun 5327   ` cfv 5333  (class class class)co 6028   ^pm cpm 6861   CCcc 8073  TopOnctopon 14804   ~~> tclm 14981

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pm 6863  df-top 14792  df-topon 14805  df-lm 14984

This theorem is referenced by:  lmss  15040