Theorem limcresi 15340

Description: Any limit of F is also a limit of the restriction of F. (Contributed by Mario Carneiro, 28-Dec-2016.)

Assertion

Ref	Expression
limcresi	|- ( F lim CC B ) C_ ( ( F |` C ) lim CC B )

Proof of Theorem limcresi

Dummy variables d e u x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limcrcl 15332	. . . . . . 7 |- ( x e. ( F lim CC B ) -> ( F : dom F --> CC /\ dom F C_ CC /\ B e. CC ) )
2	1	simp1d 1033	. . . . . 6 |- ( x e. ( F lim CC B ) -> F : dom F --> CC )
3	1	simp2d 1034	. . . . . 6 |- ( x e. ( F lim CC B ) -> dom F C_ CC )
4	1	simp3d 1035	. . . . . 6 |- ( x e. ( F lim CC B ) -> B e. CC )
5	2, 3, 4	ellimc3ap 15335	. . . . 5 |- ( x e. ( F lim CC B ) -> ( x e. ( F lim CC B ) <-> ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. u e. dom F ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) ) ) )
6	5	ibi 176	. . . 4 |- ( x e. ( F lim CC B ) -> ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. u e. dom F ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) ) )
7		inss1 3424	. . . . . . . . 9 |- ( dom F i^i C ) C_ dom F
8		ssralv 3288	. . . . . . . . 9 |- ( ( dom F i^i C ) C_ dom F -> ( A. u e. dom F ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) -> A. u e. ( dom F i^i C ) ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) ) )
9	7, 8	ax-mp 5	. . . . . . . 8 |- ( A. u e. dom F ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) -> A. u e. ( dom F i^i C ) ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) )
10		elinel2 3391	. . . . . . . . . . . . . . 15 |- ( u e. ( dom F i^i C ) -> u e. C )
11		fvres 5651	. . . . . . . . . . . . . . 15 |- ( u e. C -> ( ( F |` C ) ` u ) = ( F ` u ) )
12	10, 11	syl 14	. . . . . . . . . . . . . 14 |- ( u e. ( dom F i^i C ) -> ( ( F |` C ) ` u ) = ( F ` u ) )
13	12	adantl 277	. . . . . . . . . . . . 13 |- ( ( x e. ( F lim CC B ) /\ u e. ( dom F i^i C ) ) -> ( ( F |` C ) ` u ) = ( F ` u ) )
14	13	fvoveq1d 6023	. . . . . . . . . . . 12 |- ( ( x e. ( F lim CC B ) /\ u e. ( dom F i^i C ) ) -> ( abs ` ( ( ( F |` C ) ` u ) - x ) ) = ( abs ` ( ( F ` u ) - x ) ) )
15	14	breq1d 4093	. . . . . . . . . . 11 |- ( ( x e. ( F lim CC B ) /\ u e. ( dom F i^i C ) ) -> ( ( abs ` ( ( ( F |` C ) ` u ) - x ) ) < e <-> ( abs ` ( ( F ` u ) - x ) ) < e ) )
16	15	imbi2d 230	. . . . . . . . . 10 |- ( ( x e. ( F lim CC B ) /\ u e. ( dom F i^i C ) ) -> ( ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( ( F |` C ) ` u ) - x ) ) < e ) <-> ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) ) )
17	16	biimprd 158	. . . . . . . . 9 |- ( ( x e. ( F lim CC B ) /\ u e. ( dom F i^i C ) ) -> ( ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) -> ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( ( F |` C ) ` u ) - x ) ) < e ) ) )
18	17	ralimdva 2597	. . . . . . . 8 |- ( x e. ( F lim CC B ) -> ( A. u e. ( dom F i^i C ) ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) -> A. u e. ( dom F i^i C ) ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( ( F |` C ) ` u ) - x ) ) < e ) ) )
19	9, 18	syl5 32	. . . . . . 7 |- ( x e. ( F lim CC B ) -> ( A. u e. dom F ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) -> A. u e. ( dom F i^i C ) ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( ( F |` C ) ` u ) - x ) ) < e ) ) )
20	19	reximdv 2631	. . . . . 6 |- ( x e. ( F lim CC B ) -> ( E. d e. RR+ A. u e. dom F ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) -> E. d e. RR+ A. u e. ( dom F i^i C ) ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( ( F |` C ) ` u ) - x ) ) < e ) ) )
21	20	ralimdv 2598	. . . . 5 |- ( x e. ( F lim CC B ) -> ( A. e e. RR+ E. d e. RR+ A. u e. dom F ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) -> A. e e. RR+ E. d e. RR+ A. u e. ( dom F i^i C ) ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( ( F |` C ) ` u ) - x ) ) < e ) ) )
22	21	anim2d 337	. . . 4 |- ( x e. ( F lim CC B ) -> ( ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. u e. dom F ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( F ` u ) - x ) ) < e ) ) -> ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. u e. ( dom F i^i C ) ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( ( F |` C ) ` u ) - x ) ) < e ) ) ) )
23	6, 22	mpd 13	. . 3 |- ( x e. ( F lim CC B ) -> ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. u e. ( dom F i^i C ) ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( ( F |` C ) ` u ) - x ) ) < e ) ) )
24		fresin 5504	. . . . 5 |- ( F : dom F --> CC -> ( F |` C ) : ( dom F i^i C ) --> CC )
25	2, 24	syl 14	. . . 4 |- ( x e. ( F lim CC B ) -> ( F |` C ) : ( dom F i^i C ) --> CC )
26	7, 3	sstrid 3235	. . . 4 |- ( x e. ( F lim CC B ) -> ( dom F i^i C ) C_ CC )
27	25, 26, 4	ellimc3ap 15335	. . 3 |- ( x e. ( F lim CC B ) -> ( x e. ( ( F |` C ) lim CC B ) <-> ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. u e. ( dom F i^i C ) ( ( u # B /\ ( abs ` ( u - B ) ) < d ) -> ( abs ` ( ( ( F |` C ) ` u ) - x ) ) < e ) ) ) )
28	23, 27	mpbird 167	. 2 |- ( x e. ( F lim CC B ) -> x e. ( ( F |` C ) lim CC B ) )
29	28	ssriv 3228	1 |- ( F lim CC B ) C_ ( ( F |` C ) lim CC B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   i^i cin 3196   C_ wss 3197   class class class wbr 4083   dom cdm 4719   |` cres 4721   -->wf 5314   ` cfv 5318  (class class class)co 6001   CCcc 7997   < clt 8181   - cmin 8317   # cap 8728   RR+crp 9849   abscabs 11508   lim CC climc 15328

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 8090

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-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-br 4084  df-opab 4146  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-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pm 6798  df-limced 15330

This theorem is referenced by:  dvidlemap  15365  dvidrelem  15366  dvidsslem  15367  dvcnp2cntop  15373  dvcoapbr  15381