Theorem limccl 15683

Description: Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)

Assertion

Ref	Expression
limccl	|- ( F lim CC B ) C_ CC

Proof of Theorem limccl

Dummy variables d e f x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . 4 |- ( w e. ( F lim CC B ) -> w e. ( F lim CC B ) )
2		df-limced 15680	. . . . . 6 |- lim CC = ( f e. ( CC ^pm CC ) , x e. CC |-> { y e. CC | ( ( f : dom f --> CC /\ dom f C_ CC ) /\ ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom f ( ( z # x /\ ( abs ` ( z - x ) ) < d ) -> ( abs ` ( ( f ` z ) - y ) ) < e ) ) ) } )
3	2	elmpocl1 6275	. . . . 5 |- ( w e. ( F lim CC B ) -> F e. ( CC ^pm CC ) )
4		limcrcl 15682	. . . . . 6 |- ( w e. ( F lim CC B ) -> ( F : dom F --> CC /\ dom F C_ CC /\ B e. CC ) )
5	4	simp3d 1042	. . . . 5 |- ( w e. ( F lim CC B ) -> B e. CC )
6		cnex 8293	. . . . . . 7 |- CC e. _V
7	6	rabex 4275	. . . . . 6 |- { y e. CC | ( ( F : dom F --> CC /\ dom F C_ CC ) /\ ( B e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) ) } e. _V
8	7	a1i 9	. . . . 5 |- ( w e. ( F lim CC B ) -> { y e. CC | ( ( F : dom F --> CC /\ dom F C_ CC ) /\ ( B e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) ) } e. _V )
9		simpl 109	. . . . . . . . . 10 |- ( ( f = F /\ x = B ) -> f = F )
10	9	dmeqd 4978	. . . . . . . . . 10 |- ( ( f = F /\ x = B ) -> dom f = dom F )
11	9, 10	feq12d 5518	. . . . . . . . 9 |- ( ( f = F /\ x = B ) -> ( f : dom f --> CC <-> F : dom F --> CC ) )
12	10	sseq1d 3277	. . . . . . . . 9 |- ( ( f = F /\ x = B ) -> ( dom f C_ CC <-> dom F C_ CC ) )
13	11, 12	anbi12d 477	. . . . . . . 8 |- ( ( f = F /\ x = B ) -> ( ( f : dom f --> CC /\ dom f C_ CC ) <-> ( F : dom F --> CC /\ dom F C_ CC ) ) )
14		simpr 110	. . . . . . . . . 10 |- ( ( f = F /\ x = B ) -> x = B )
15	14	eleq1d 2307	. . . . . . . . 9 |- ( ( f = F /\ x = B ) -> ( x e. CC <-> B e. CC ) )
16	14	breq2d 4137	. . . . . . . . . . . . . 14 |- ( ( f = F /\ x = B ) -> ( z # x <-> z # B ) )
17	14	oveq2d 6091	. . . . . . . . . . . . . . . 16 |- ( ( f = F /\ x = B ) -> ( z - x ) = ( z - B ) )
18	17	fveq2d 5694	. . . . . . . . . . . . . . 15 |- ( ( f = F /\ x = B ) -> ( abs ` ( z - x ) ) = ( abs ` ( z - B ) ) )
19	18	breq1d 4135	. . . . . . . . . . . . . 14 |- ( ( f = F /\ x = B ) -> ( ( abs ` ( z - x ) ) < d <-> ( abs ` ( z - B ) ) < d ) )
20	16, 19	anbi12d 477	. . . . . . . . . . . . 13 |- ( ( f = F /\ x = B ) -> ( ( z # x /\ ( abs ` ( z - x ) ) < d ) <-> ( z # B /\ ( abs ` ( z - B ) ) < d ) ) )
21	9	fveq1d 5692	. . . . . . . . . . . . . . 15 |- ( ( f = F /\ x = B ) -> ( f ` z ) = ( F ` z ) )
22	21	fvoveq1d 6097	. . . . . . . . . . . . . 14 |- ( ( f = F /\ x = B ) -> ( abs ` ( ( f ` z ) - y ) ) = ( abs ` ( ( F ` z ) - y ) ) )
23	22	breq1d 4135	. . . . . . . . . . . . 13 |- ( ( f = F /\ x = B ) -> ( ( abs ` ( ( f ` z ) - y ) ) < e <-> ( abs ` ( ( F ` z ) - y ) ) < e ) )
24	20, 23	imbi12d 234	. . . . . . . . . . . 12 |- ( ( f = F /\ x = B ) -> ( ( ( z # x /\ ( abs ` ( z - x ) ) < d ) -> ( abs ` ( ( f ` z ) - y ) ) < e ) <-> ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) )
25	10, 24	raleqbidv 2765	. . . . . . . . . . 11 |- ( ( f = F /\ x = B ) -> ( A. z e. dom f ( ( z # x /\ ( abs ` ( z - x ) ) < d ) -> ( abs ` ( ( f ` z ) - y ) ) < e ) <-> A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) )
26	25	rexbidv 2551	. . . . . . . . . 10 |- ( ( f = F /\ x = B ) -> ( E. d e. RR+ A. z e. dom f ( ( z # x /\ ( abs ` ( z - x ) ) < d ) -> ( abs ` ( ( f ` z ) - y ) ) < e ) <-> E. d e. RR+ A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) )
27	26	ralbidv 2550	. . . . . . . . 9 |- ( ( f = F /\ x = B ) -> ( A. e e. RR+ E. d e. RR+ A. z e. dom f ( ( z # x /\ ( abs ` ( z - x ) ) < d ) -> ( abs ` ( ( f ` z ) - y ) ) < e ) <-> A. e e. RR+ E. d e. RR+ A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) )
28	15, 27	anbi12d 477	. . . . . . . 8 |- ( ( f = F /\ x = B ) -> ( ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom f ( ( z # x /\ ( abs ` ( z - x ) ) < d ) -> ( abs ` ( ( f ` z ) - y ) ) < e ) ) <-> ( B e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) ) )
29	13, 28	anbi12d 477	. . . . . . 7 |- ( ( f = F /\ x = B ) -> ( ( ( f : dom f --> CC /\ dom f C_ CC ) /\ ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom f ( ( z # x /\ ( abs ` ( z - x ) ) < d ) -> ( abs ` ( ( f ` z ) - y ) ) < e ) ) ) <-> ( ( F : dom F --> CC /\ dom F C_ CC ) /\ ( B e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) ) ) )
30	29	rabbidv 2810	. . . . . 6 |- ( ( f = F /\ x = B ) -> { y e. CC | ( ( f : dom f --> CC /\ dom f C_ CC ) /\ ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom f ( ( z # x /\ ( abs ` ( z - x ) ) < d ) -> ( abs ` ( ( f ` z ) - y ) ) < e ) ) ) } = { y e. CC | ( ( F : dom F --> CC /\ dom F C_ CC ) /\ ( B e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) ) } )
31	30, 2	ovmpoga 6208	. . . . 5 |- ( ( F e. ( CC ^pm CC ) /\ B e. CC /\ { y e. CC | ( ( F : dom F --> CC /\ dom F C_ CC ) /\ ( B e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) ) } e. _V ) -> ( F lim CC B ) = { y e. CC | ( ( F : dom F --> CC /\ dom F C_ CC ) /\ ( B e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) ) } )
32	3, 5, 8, 31	syl3anc 1278	. . . 4 |- ( w e. ( F lim CC B ) -> ( F lim CC B ) = { y e. CC | ( ( F : dom F --> CC /\ dom F C_ CC ) /\ ( B e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) ) } )
33	1, 32	eleqtrd 2317	. . 3 |- ( w e. ( F lim CC B ) -> w e. { y e. CC | ( ( F : dom F --> CC /\ dom F C_ CC ) /\ ( B e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) ) } )
34		elrabi 2979	. . 3 |- ( w e. { y e. CC | ( ( F : dom F --> CC /\ dom F C_ CC ) /\ ( B e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom F ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - y ) ) < e ) ) ) } -> w e. CC )
35	33, 34	syl 14	. 2 |- ( w e. ( F lim CC B ) -> w e. CC )
36	35	ssriv 3252	1 |- ( F lim CC B ) C_ CC

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   _Vcvv 2821   C_ wss 3220   class class class wbr 4125   dom cdm 4769   -->wf 5368   ` cfv 5372  (class class class)co 6075   ^pm cpm 6913   CCcc 8167   < clt 8350   - cmin 8487   # cap 8899   RR+crp 10033   abscabs 11741   lim CC climc 15678

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pm 6915  df-limced 15680

This theorem is referenced by:  reldvg  15703  dvfvalap  15705  dvcl  15707