Theorem lmfss 14933

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 14932	. . 3 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> F e. ( X ^pm CC ) )
2		toponmax 14714	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
3		cnex 8134	. . . . 5 |- CC e. _V
4		elpmg 6819	. . . . 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 2200   _Vcvv 2799   C_ wss 3197   class class class wbr 4083   X. cxp 4717   Fun wfun 5312   ` cfv 5318  (class class class)co 6007   ^pm cpm 6804   CCcc 8008  TopOnctopon 14699   ~~> tclm 14876

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pm 6806  df-top 14687  df-topon 14700  df-lm 14879

This theorem is referenced by:  lmss  14935