Theorem limcmpted 14670

Description: Express the limit operator for a function defined by a mapping, via epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.)

Hypotheses

Ref	Expression
limcmpted.a	|- ( ph -> A C_ CC )
limcmpted.b	|- ( ph -> B e. CC )
limcmpted.f	|- ( ( ph /\ z e. A ) -> D e. CC )

Assertion

Ref	Expression
limcmpted	|- ( ph -> ( C e. ( ( z e. A |-> D ) lim CC B ) <-> ( C e. CC /\ A. x e. RR+ E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( D - C ) ) < x ) ) ) )

Distinct variable groups:   x, A, y, z   x, B, y, z   x, C, y, z   x, D, y   ph, x, y, z

Allowed substitution hint:   D( z)

Proof of Theorem limcmpted

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2332	. . . . . 6 |- F/_ w D
2		nfcsb1v 3109	. . . . . 6 |- F/_ z [_ w / z ]_ D
3		csbeq1a 3085	. . . . . 6 |- ( z = w -> D = [_ w / z ]_ D )
4	1, 2, 3	cbvmpt 4120	. . . . 5 |- ( z e. A |-> D ) = ( w e. A |-> [_ w / z ]_ D )
5	4	a1i 9	. . . 4 |- ( ph -> ( z e. A |-> D ) = ( w e. A |-> [_ w / z ]_ D ) )
6	5	oveq1d 5919	. . 3 |- ( ph -> ( ( z e. A |-> D ) lim CC B ) = ( ( w e. A |-> [_ w / z ]_ D ) lim CC B ) )
7	6	eleq2d 2259	. 2 |- ( ph -> ( C e. ( ( z e. A |-> D ) lim CC B ) <-> C e. ( ( w e. A |-> [_ w / z ]_ D ) lim CC B ) ) )
8		limcmpted.f	. . . . 5 |- ( ( ph /\ z e. A ) -> D e. CC )
9	8	fmpttd 5699	. . . 4 |- ( ph -> ( z e. A |-> D ) : A --> CC )
10	4	feq1i 5384	. . . 4 |- ( ( z e. A |-> D ) : A --> CC <-> ( w e. A |-> [_ w / z ]_ D ) : A --> CC )
11	9, 10	sylib 122	. . 3 |- ( ph -> ( w e. A |-> [_ w / z ]_ D ) : A --> CC )
12		limcmpted.a	. . 3 |- ( ph -> A C_ CC )
13		limcmpted.b	. . 3 |- ( ph -> B e. CC )
14		nfcv 2332	. . . 4 |- F/_ z A
15	14, 2	nfmpt 4117	. . 3 |- F/_ z ( w e. A |-> [_ w / z ]_ D )
16	11, 12, 13, 15	ellimc3apf 14667	. 2 |- ( ph -> ( C e. ( ( w e. A |-> [_ w / z ]_ D ) lim CC B ) <-> ( C e. CC /\ A. x e. RR+ E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( ( ( w e. A |-> [_ w / z ]_ D ) ` z ) - C ) ) < x ) ) ) )
17		eqid 2189	. . . . . . . . . 10 |- ( w e. A |-> [_ w / z ]_ D ) = ( w e. A |-> [_ w / z ]_ D )
18		eqcom 2191	. . . . . . . . . . 11 |- ( z = w <-> w = z )
19		eqcom 2191	. . . . . . . . . . 11 |- ( D = [_ w / z ]_ D <-> [_ w / z ]_ D = D )
20	3, 18, 19	3imtr3i 200	. . . . . . . . . 10 |- ( w = z -> [_ w / z ]_ D = D )
21		simpr 110	. . . . . . . . . 10 |- ( ( ph /\ z e. A ) -> z e. A )
22	17, 20, 21, 8	fvmptd3 5637	. . . . . . . . 9 |- ( ( ph /\ z e. A ) -> ( ( w e. A |-> [_ w / z ]_ D ) ` z ) = D )
23	22	fvoveq1d 5926	. . . . . . . 8 |- ( ( ph /\ z e. A ) -> ( abs ` ( ( ( w e. A |-> [_ w / z ]_ D ) ` z ) - C ) ) = ( abs ` ( D - C ) ) )
24	23	breq1d 4035	. . . . . . 7 |- ( ( ph /\ z e. A ) -> ( ( abs ` ( ( ( w e. A |-> [_ w / z ]_ D ) ` z ) - C ) ) < x <-> ( abs ` ( D - C ) ) < x ) )
25	24	imbi2d 230	. . . . . 6 |- ( ( ph /\ z e. A ) -> ( ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( ( ( w e. A |-> [_ w / z ]_ D ) ` z ) - C ) ) < x ) <-> ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( D - C ) ) < x ) ) )
26	25	ralbidva 2486	. . . . 5 |- ( ph -> ( A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( ( ( w e. A |-> [_ w / z ]_ D ) ` z ) - C ) ) < x ) <-> A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( D - C ) ) < x ) ) )
27	26	rexbidv 2491	. . . 4 |- ( ph -> ( E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( ( ( w e. A |-> [_ w / z ]_ D ) ` z ) - C ) ) < x ) <-> E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( D - C ) ) < x ) ) )
28	27	ralbidv 2490	. . 3 |- ( ph -> ( A. x e. RR+ E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( ( ( w e. A |-> [_ w / z ]_ D ) ` z ) - C ) ) < x ) <-> A. x e. RR+ E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( D - C ) ) < x ) ) )
29	28	anbi2d 464	. 2 |- ( ph -> ( ( C e. CC /\ A. x e. RR+ E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( ( ( w e. A |-> [_ w / z ]_ D ) ` z ) - C ) ) < x ) ) <-> ( C e. CC /\ A. x e. RR+ E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( D - C ) ) < x ) ) ) )
30	7, 16, 29	3bitrd 214	1 |- ( ph -> ( C e. ( ( z e. A |-> D ) lim CC B ) <-> ( C e. CC /\ A. x e. RR+ E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( D - C ) ) < x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   A.wral 2468   E.wrex 2469   [_csb 3076   C_ wss 3148   class class class wbr 4025   |-> cmpt 4086   -->wf 5238   ` cfv 5242  (class class class)co 5904   CCcc 7849   < clt 8033   - cmin 8169   # cap 8579   RR+crp 9695   abscabs 11053   lim CC climc 14661

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-cnex 7942

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3150  df-un 3152  df-in 3154  df-ss 3161  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-br 4026  df-opab 4087  df-mpt 4088  df-id 4318  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-fv 5250  df-ov 5907  df-oprab 5908  df-mpo 5909  df-pm 6685  df-limced 14663

This theorem is referenced by:  limccnp2cntop  14684  limccoap  14685