Theorem limcmpted 14272

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 2319	. . . . . 6 |- F/_ w D
2		nfcsb1v 3092	. . . . . 6 |- F/_ z [_ w / z ]_ D
3		csbeq1a 3068	. . . . . 6 |- ( z = w -> D = [_ w / z ]_ D )
4	1, 2, 3	cbvmpt 4100	. . . . 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 5893	. . 3 |- ( ph -> ( ( z e. A |-> D ) lim CC B ) = ( ( w e. A |-> [_ w / z ]_ D ) lim CC B ) )
7	6	eleq2d 2247	. 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 5674	. . . 4 |- ( ph -> ( z e. A |-> D ) : A --> CC )
10	4	feq1i 5360	. . . 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 2319	. . . 4 |- F/_ z A
15	14, 2	nfmpt 4097	. . 3 |- F/_ z ( w e. A |-> [_ w / z ]_ D )
16	11, 12, 13, 15	ellimc3apf 14269	. 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 2177	. . . . . . . . . 10 |- ( w e. A |-> [_ w / z ]_ D ) = ( w e. A |-> [_ w / z ]_ D )
18		eqcom 2179	. . . . . . . . . . 11 |- ( z = w <-> w = z )
19		eqcom 2179	. . . . . . . . . . 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 5612	. . . . . . . . 9 |- ( ( ph /\ z e. A ) -> ( ( w e. A |-> [_ w / z ]_ D ) ` z ) = D )
23	22	fvoveq1d 5900	. . . . . . . 8 |- ( ( ph /\ z e. A ) -> ( abs ` ( ( ( w e. A |-> [_ w / z ]_ D ) ` z ) - C ) ) = ( abs ` ( D - C ) ) )
24	23	breq1d 4015	. . . . . . 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 2473	. . . . 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 2478	. . . 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 2477	. . 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 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   [_csb 3059   C_ wss 3131   class class class wbr 4005   |-> cmpt 4066   -->wf 5214   ` cfv 5218  (class class class)co 5878   CCcc 7812   < clt 7995   - cmin 8131   # cap 8541   RR+crp 9656   abscabs 11009   lim CC climc 14263

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pm 6654  df-limced 14265

This theorem is referenced by:  limccnp2cntop  14286  limccoap  14287