Theorem lmfss 13038

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 13037	. . 3 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> F e. ( X ^pm CC ) )
2		toponmax 12817	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
3		cnex 7898	. . . . 5 |- CC e. _V
4		elpmg 6642	. . . . 5 |- ( ( X e. J /\ CC e. _V ) -> ( F e. ( X ^pm CC ) <-> ( Fun F /\ F C_ ( CC X. X ) ) ) )
5	2, 3, 4	sylancl 411	. . . 4 |- ( J e. (TopOn ` X ) -> ( F e. ( X ^pm CC ) <-> ( Fun F /\ F C_ ( CC X. X ) ) ) )
6	5	adantr 274	. . 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 146	. 2 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> ( Fun F /\ F C_ ( CC X. X ) ) )
8	7	simprd 113	1 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> F C_ ( CC X. X ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   _Vcvv 2730   C_ wss 3121   class class class wbr 3989   X. cxp 4609   Fun wfun 5192   ` cfv 5198  (class class class)co 5853   ^pm cpm 6627   CCcc 7772  TopOnctopon 12802   ~~> tclm 12981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pm 6629  df-top 12790  df-topon 12803  df-lm 12984

This theorem is referenced by:  lmss  13040