Theorem limcresi 15477

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 15469	. . . . . . 7 |- ( x e. ( F lim CC B ) -> ( F : dom F --> CC /\ dom F C_ CC /\ B e. CC ) )
2	1	simp1d 1036	. . . . . 6 |- ( x e. ( F lim CC B ) -> F : dom F --> CC )
3	1	simp2d 1037	. . . . . 6 |- ( x e. ( F lim CC B ) -> dom F C_ CC )
4	1	simp3d 1038	. . . . . 6 |- ( x e. ( F lim CC B ) -> B e. CC )
5	2, 3, 4	ellimc3ap 15472	. . . . 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 3429	. . . . . . . . 9 |- ( dom F i^i C ) C_ dom F
8		ssralv 3292	. . . . . . . . 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 3396	. . . . . . . . . . . . . . 15 |- ( u e. ( dom F i^i C ) -> u e. C )
11		fvres 5672	. . . . . . . . . . . . . . 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 6050	. . . . . . . . . . . 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 4103	. . . . . . . . . . 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 2600	. . . . . . . 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 2634	. . . . . 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 2601	. . . . 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 5523	. . . . 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 3239	. . . 4 |- ( x e. ( F lim CC B ) -> ( dom F i^i C ) C_ CC )
27	25, 26, 4	ellimc3ap 15472	. . 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 3232	1 |- ( F lim CC B ) C_ ( ( F |` C ) lim CC B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   i^i cin 3200   C_ wss 3201   class class class wbr 4093   dom cdm 4731   |` cres 4733   -->wf 5329   ` cfv 5333  (class class class)co 6028   CCcc 8090   < clt 8273   - cmin 8409   # cap 8820   RR+crp 9949   abscabs 11637   lim CC climc 15465

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 8183

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-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-br 4094  df-opab 4156  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-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pm 6863  df-limced 15467

This theorem is referenced by:  dvidlemap  15502  dvidrelem  15503  dvidsslem  15504  dvcnp2cntop  15510  dvcoapbr  15518