Theorem limcdifap 15686

Description: It suffices to consider functions which are not defined at B to define the limit of a function. In particular, the value of the original function F at B does not affect the limit of F. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)

Hypotheses

Ref	Expression
limccl.f	|- ( ph -> F : A --> CC )
limcdifap.a	|- ( ph -> A C_ CC )

Assertion

Ref	Expression
limcdifap	|- ( ph -> ( F lim CC B ) = ( ( F |` { x e. A | x # B } ) lim CC B ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   F( x)

Proof of Theorem limcdifap

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

Step	Hyp	Ref	Expression
1		limcrcl 15682	. . . . 5 |- ( u e. ( F lim CC B ) -> ( F : dom F --> CC /\ dom F C_ CC /\ B e. CC ) )
2	1	simp3d 1042	. . . 4 |- ( u e. ( F lim CC B ) -> B e. CC )
3	2	a1i 9	. . 3 |- ( ph -> ( u e. ( F lim CC B ) -> B e. CC ) )
4		limcrcl 15682	. . . . 5 |- ( u e. ( ( F |` { x e. A | x # B } ) lim CC B ) -> ( ( F |` { x e. A | x # B } ) : dom ( F |` { x e. A | x # B } ) --> CC /\ dom ( F |` { x e. A | x # B } ) C_ CC /\ B e. CC ) )
5	4	simp3d 1042	. . . 4 |- ( u e. ( ( F |` { x e. A | x # B } ) lim CC B ) -> B e. CC )
6	5	a1i 9	. . 3 |- ( ph -> ( u e. ( ( F |` { x e. A | x # B } ) lim CC B ) -> B e. CC ) )
7		breq1 4128	. . . . . . . . . . . . . . . . 17 |- ( x = z -> ( x # B <-> z # B ) )
8		simplr 533	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ B e. CC ) /\ z e. A ) /\ z # B ) -> z e. A )
9		simpr 110	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ B e. CC ) /\ z e. A ) /\ z # B ) -> z # B )
10	7, 8, 9	elrabd 2984	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ B e. CC ) /\ z e. A ) /\ z # B ) -> z e. { x e. A | x # B } )
11		fvres 5714	. . . . . . . . . . . . . . . . 17 |- ( z e. { x e. A | x # B } -> ( ( F |` { x e. A | x # B } ) ` z ) = ( F ` z ) )
12	11	eqcomd 2244	. . . . . . . . . . . . . . . 16 |- ( z e. { x e. A | x # B } -> ( F ` z ) = ( ( F |` { x e. A | x # B } ) ` z ) )
13	10, 12	syl 14	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ B e. CC ) /\ z e. A ) /\ z # B ) -> ( F ` z ) = ( ( F |` { x e. A | x # B } ) ` z ) )
14	13	fvoveq1d 6097	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ B e. CC ) /\ z e. A ) /\ z # B ) -> ( abs ` ( ( F ` z ) - u ) ) = ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) )
15	14	breq1d 4135	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ B e. CC ) /\ z e. A ) /\ z # B ) -> ( ( abs ` ( ( F ` z ) - u ) ) < e <-> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) )
16	15	imbi2d 230	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ B e. CC ) /\ z e. A ) /\ z # B ) -> ( ( ( abs ` ( z - B ) ) < d -> ( abs ` ( ( F ` z ) - u ) ) < e ) <-> ( ( abs ` ( z - B ) ) < d -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) )
17	16	pm5.74da 447	. . . . . . . . . . 11 |- ( ( ( ph /\ B e. CC ) /\ z e. A ) -> ( ( z # B -> ( ( abs ` ( z - B ) ) < d -> ( abs ` ( ( F ` z ) - u ) ) < e ) ) <-> ( z # B -> ( ( abs ` ( z - B ) ) < d -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) ) )
18		impexp 263	. . . . . . . . . . 11 |- ( ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - u ) ) < e ) <-> ( z # B -> ( ( abs ` ( z - B ) ) < d -> ( abs ` ( ( F ` z ) - u ) ) < e ) ) )
19		impexp 263	. . . . . . . . . . . . 13 |- ( ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) <-> ( z # B -> ( ( abs ` ( z - B ) ) < d -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) )
20	19	imbi2i 226	. . . . . . . . . . . 12 |- ( ( z # B -> ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) <-> ( z # B -> ( z # B -> ( ( abs ` ( z - B ) ) < d -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) ) )
21		pm5.4 249	. . . . . . . . . . . 12 |- ( ( z # B -> ( z # B -> ( ( abs ` ( z - B ) ) < d -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) ) <-> ( z # B -> ( ( abs ` ( z - B ) ) < d -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) )
22	20, 21	bitri 184	. . . . . . . . . . 11 |- ( ( z # B -> ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) <-> ( z # B -> ( ( abs ` ( z - B ) ) < d -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) )
23	17, 18, 22	3bitr4g 223	. . . . . . . . . 10 |- ( ( ( ph /\ B e. CC ) /\ z e. A ) -> ( ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - u ) ) < e ) <-> ( z # B -> ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) ) )
24	23	ralbidva 2546	. . . . . . . . 9 |- ( ( ph /\ B e. CC ) -> ( A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - u ) ) < e ) <-> A. z e. A ( z # B -> ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) ) )
25	7	ralrab 2987	. . . . . . . . 9 |- ( A. z e. { x e. A | x # B } ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) <-> A. z e. A ( z # B -> ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) )
26	24, 25	bitr4di 198	. . . . . . . 8 |- ( ( ph /\ B e. CC ) -> ( A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - u ) ) < e ) <-> A. z e. { x e. A | x # B } ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) )
27	26	rexbidv 2551	. . . . . . 7 |- ( ( ph /\ B e. CC ) -> ( E. d e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - u ) ) < e ) <-> E. d e. RR+ A. z e. { x e. A | x # B } ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) )
28	27	ralbidv 2550	. . . . . 6 |- ( ( ph /\ B e. CC ) -> ( A. e e. RR+ E. d e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - u ) ) < e ) <-> A. e e. RR+ E. d e. RR+ A. z e. { x e. A | x # B } ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) )
29	28	anbi2d 468	. . . . 5 |- ( ( ph /\ B e. CC ) -> ( ( u e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - u ) ) < e ) ) <-> ( u e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. { x e. A | x # B } ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) ) )
30		limccl.f	. . . . . . 7 |- ( ph -> F : A --> CC )
31	30	adantr 276	. . . . . 6 |- ( ( ph /\ B e. CC ) -> F : A --> CC )
32		limcdifap.a	. . . . . . 7 |- ( ph -> A C_ CC )
33	32	adantr 276	. . . . . 6 |- ( ( ph /\ B e. CC ) -> A C_ CC )
34		simpr 110	. . . . . 6 |- ( ( ph /\ B e. CC ) -> B e. CC )
35	31, 33, 34	ellimc3ap 15685	. . . . 5 |- ( ( ph /\ B e. CC ) -> ( u e. ( F lim CC B ) <-> ( u e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( F ` z ) - u ) ) < e ) ) ) )
36		ssrab2 3333	. . . . . . 7 |- { x e. A | x # B } C_ A
37		fssres 5560	. . . . . . 7 |- ( ( F : A --> CC /\ { x e. A | x # B } C_ A ) -> ( F |` { x e. A | x # B } ) : { x e. A | x # B } --> CC )
38	31, 36, 37	sylancl 417	. . . . . 6 |- ( ( ph /\ B e. CC ) -> ( F |` { x e. A | x # B } ) : { x e. A | x # B } --> CC )
39	36, 33	sstrid 3259	. . . . . 6 |- ( ( ph /\ B e. CC ) -> { x e. A | x # B } C_ CC )
40	38, 39, 34	ellimc3ap 15685	. . . . 5 |- ( ( ph /\ B e. CC ) -> ( u e. ( ( F |` { x e. A | x # B } ) lim CC B ) <-> ( u e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. { x e. A | x # B } ( ( z # B /\ ( abs ` ( z - B ) ) < d ) -> ( abs ` ( ( ( F |` { x e. A | x # B } ) ` z ) - u ) ) < e ) ) ) )
41	29, 35, 40	3bitr4d 220	. . . 4 |- ( ( ph /\ B e. CC ) -> ( u e. ( F lim CC B ) <-> u e. ( ( F |` { x e. A | x # B } ) lim CC B ) ) )
42	41	ex 115	. . 3 |- ( ph -> ( B e. CC -> ( u e. ( F lim CC B ) <-> u e. ( ( F |` { x e. A | x # B } ) lim CC B ) ) ) )
43	3, 6, 42	pm5.21ndd 717	. 2 |- ( ph -> ( u e. ( F lim CC B ) <-> u e. ( ( F |` { x e. A | x # B } ) lim CC B ) ) )
44	43	eqrdv 2236	1 |- ( ph -> ( F lim CC B ) = ( ( F |` { x e. A | x # B } ) lim CC B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   C_ wss 3220   class class class wbr 4125   dom cdm 4769   |` cres 4771   -->wf 5368   ` cfv 5372  (class class class)co 6075   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-res 4781  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:  dvcnp2cntop  15723  dvmulxxbr  15726  dvrecap  15737