Theorem lmfss 14716

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 14715	. . 3 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> F e. ( X ^pm CC ) )
2		toponmax 14497	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
3		cnex 8049	. . . . 5 |- CC e. _V
4		elpmg 6751	. . . . 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 2176   _Vcvv 2772   C_ wss 3166   class class class wbr 4044   X. cxp 4673   Fun wfun 5265   ` cfv 5271  (class class class)co 5944   ^pm cpm 6736   CCcc 7923  TopOnctopon 14482   ~~> tclm 14659

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-pm 6738  df-top 14470  df-topon 14483  df-lm 14662

This theorem is referenced by:  lmss  14718