Theorem cncffvrn 8535

Description: Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.)

Assertion

Ref	Expression
cncffvrn	|- (((A C_ CC /\ B C_ CC /\ C C_ CC) /\ A.x e. A (F` x) e. C) -> (F e. (A-cn->B) -> F e. (A-cn->C)))

Distinct variable groups:   x,A   x,C   x,F

Proof of Theorem cncffvrn

Step	Hyp	Ref	Expression
1		fnfvrnss 4803	. . . . . . 7 |- ((F Fn A /\ A.x e. A (F` x) e. C) -> ran F C_ C)
2		ffn 4562	. . . . . . 7 |- (F:A-->B -> F Fn A)
3	1, 2	sylan 497	. . . . . 6 |- ((F:A-->B /\ A.x e. A (F` x) e. C) -> ran F C_ C)
4		fss 4571	. . . . . . 7 |- ((F:A-->ran F /\ ran F C_ C) -> F:A-->C)
5		dffn3 4570	. . . . . . . 8 |- (F Fn A <-> F:A-->ran F)
6	2, 5	sylib 215	. . . . . . 7 |- (F:A-->B -> F:A-->ran F)
7	4, 6	sylan 497	. . . . . 6 |- ((F:A-->B /\ ran F C_ C) -> F:A-->C)
8	3, 7	syldan 516	. . . . 5 |- ((F:A-->B /\ A.x e. A (F` x) e. C) -> F:A-->C)
9	8	expcom 403	. . . 4 |- (A.x e. A (F` x) e. C -> (F:A-->B -> F:A-->C))
10	9	anim1d 619	. . 3 |- (A.x e. A (F` x) e. C -> ((F:A-->B /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)) -> (F:A-->C /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))))
11	10	adantl 424	. 2 |- (((A C_ CC /\ B C_ CC /\ C C_ CC) /\ A.x e. A (F` x) e. C) -> ((F:A-->B /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)) -> (F:A-->C /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))))
12		elcncf 8527	. . . 4 |- ((A C_ CC /\ B C_ CC) -> (F e. (A-cn->B) <-> (F:A-->B /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))))
13	12	3adant3 896	. . 3 |- ((A C_ CC /\ B C_ CC /\ C C_ CC) -> (F e. (A-cn->B) <-> (F:A-->B /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))))
14	13	adantr 425	. 2 |- (((A C_ CC /\ B C_ CC /\ C C_ CC) /\ A.x e. A (F` x) e. C) -> (F e. (A-cn->B) <-> (F:A-->B /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))))
15		elcncf 8527	. . . 4 |- ((A C_ CC /\ C C_ CC) -> (F e. (A-cn->C) <-> (F:A-->C /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))))
16	15	3adant2 895	. . 3 |- ((A C_ CC /\ B C_ CC /\ C C_ CC) -> (F e. (A-cn->C) <-> (F:A-->C /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))))
17	16	adantr 425	. 2 |- (((A C_ CC /\ B C_ CC /\ C C_ CC) /\ A.x e. A (F` x) e. C) -> (F e. (A-cn->C) <-> (F:A-->C /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))))
18	11, 14, 17	3imtr4d 602	1 |- (((A C_ CC /\ B C_ CC /\ C C_ CC) /\ A.x e. A (F` x) e. C) -> (F e. (A-cn->B) -> F e. (A-cn->C)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593   class class class wbr 3338  ran crn 3987   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384   - cmin 6445  RR+crp 6453   < clt 6653  abscabs 8000  -cn->ccncf 8524

This theorem is referenced by:  isupivthi 8552  cncfres 15895  phtpycom 16050  phtpycolem3 16053  phtpycolem4 16054  pcocn 16076

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-qs 5323  df-ni 6152  df-nq 6190  df-np 6238  df-nr 6319  df-c 6392  df-cncf 8525

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