Theorem limcmpted 15377

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 2372	. . . . . 6 |- F/_ w D
2		nfcsb1v 3158	. . . . . 6 |- F/_ z [_ w / z ]_ D
3		csbeq1a 3134	. . . . . 6 |- ( z = w -> D = [_ w / z ]_ D )
4	1, 2, 3	cbvmpt 4182	. . . . 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 6028	. . 3 |- ( ph -> ( ( z e. A |-> D ) lim CC B ) = ( ( w e. A |-> [_ w / z ]_ D ) lim CC B ) )
7	6	eleq2d 2299	. 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 5798	. . . 4 |- ( ph -> ( z e. A |-> D ) : A --> CC )
10	4	feq1i 5472	. . . 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 2372	. . . 4 |- F/_ z A
15	14, 2	nfmpt 4179	. . 3 |- F/_ z ( w e. A |-> [_ w / z ]_ D )
16	11, 12, 13, 15	ellimc3apf 15374	. 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 2229	. . . . . . . . . 10 |- ( w e. A |-> [_ w / z ]_ D ) = ( w e. A |-> [_ w / z ]_ D )
18		eqcom 2231	. . . . . . . . . . 11 |- ( z = w <-> w = z )
19		eqcom 2231	. . . . . . . . . . 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 5736	. . . . . . . . 9 |- ( ( ph /\ z e. A ) -> ( ( w e. A |-> [_ w / z ]_ D ) ` z ) = D )
23	22	fvoveq1d 6035	. . . . . . . 8 |- ( ( ph /\ z e. A ) -> ( abs ` ( ( ( w e. A |-> [_ w / z ]_ D ) ` z ) - C ) ) = ( abs ` ( D - C ) ) )
24	23	breq1d 4096	. . . . . . 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 2526	. . . . 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 2531	. . . 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 2530	. . 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 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   [_csb 3125   C_ wss 3198   class class class wbr 4086   |-> cmpt 4148   -->wf 5320   ` cfv 5324  (class class class)co 6013   CCcc 8020   < clt 8204   - cmin 8340   # cap 8751   RR+crp 9878   abscabs 11548   lim CC climc 15368

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pm 6815  df-limced 15370

This theorem is referenced by:  limccnp2cntop  15391  limccoap  15392