Theorem limcmpted 15457

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 2375	. . . . . 6 |- F/_ w D
2		nfcsb1v 3161	. . . . . 6 |- F/_ z [_ w / z ]_ D
3		csbeq1a 3137	. . . . . 6 |- ( z = w -> D = [_ w / z ]_ D )
4	1, 2, 3	cbvmpt 4189	. . . . 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 6043	. . 3 |- ( ph -> ( ( z e. A |-> D ) lim CC B ) = ( ( w e. A |-> [_ w / z ]_ D ) lim CC B ) )
7	6	eleq2d 2301	. 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 5810	. . . 4 |- ( ph -> ( z e. A |-> D ) : A --> CC )
10	4	feq1i 5482	. . . 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 2375	. . . 4 |- F/_ z A
15	14, 2	nfmpt 4186	. . 3 |- F/_ z ( w e. A |-> [_ w / z ]_ D )
16	11, 12, 13, 15	ellimc3apf 15454	. 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 2231	. . . . . . . . . 10 |- ( w e. A |-> [_ w / z ]_ D ) = ( w e. A |-> [_ w / z ]_ D )
18		eqcom 2233	. . . . . . . . . . 11 |- ( z = w <-> w = z )
19		eqcom 2233	. . . . . . . . . . 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 5749	. . . . . . . . 9 |- ( ( ph /\ z e. A ) -> ( ( w e. A |-> [_ w / z ]_ D ) ` z ) = D )
23	22	fvoveq1d 6050	. . . . . . . 8 |- ( ( ph /\ z e. A ) -> ( abs ` ( ( ( w e. A |-> [_ w / z ]_ D ) ` z ) - C ) ) = ( abs ` ( D - C ) ) )
24	23	breq1d 4103	. . . . . . 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 2529	. . . . 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 2534	. . . 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 2533	. . 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 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   [_csb 3128   C_ wss 3201   class class class wbr 4093   |-> cmpt 4155   -->wf 5329   ` cfv 5333  (class class class)co 6028   CCcc 8073   < clt 8256   - cmin 8392   # cap 8803   RR+crp 9932   abscabs 11620   lim CC climc 15448

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 8166

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-csb 3129  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-mpt 4157  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-ima 4744  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 15450

This theorem is referenced by:  limccnp2cntop  15471  limccoap  15472