Theorem cncffvrn 7273

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

Assertion

Ref	Expression
cncffvrn	|- (((A (_ CC /\ B (_ CC /\ 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 3830	. . . . . . 7 |- ((F Fn A /\ A.x e. A (F` x) e. C) -> ran F (_ C)
2		ffn 3627	. . . . . . 7 |- (F:A-->B -> F Fn A)
3	1, 2	sylan 448	. . . . . 6 |- ((F:A-->B /\ A.x e. A (F` x) e. C) -> ran F (_ C)
4		fss 3635	. . . . . . 7 |- ((F:A-->ran F /\ ran F (_ C) -> F:A-->C)
5		fnfrn 3634	. . . . . . . 8 |- (F Fn A <-> F:A-->ran F)
6	2, 5	sylib 198	. . . . . . 7 |- (F:A-->B -> F:A-->ran F)
7	4, 6	sylan 448	. . . . . 6 |- ((F:A-->B /\ ran F (_ C) -> F:A-->C)
8	3, 7	syldan 467	. . . . 5 |- ((F:A-->B /\ A.x e. A (F` x) e. C) -> F:A-->C)
9	8	expcom 374	. . . 4 |- (A.x e. A (F` x) e. C -> (F:A-->B -> F:A-->C))
10	9	anim1d 560	. . 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 388	. 2 |- (((A (_ CC /\ B (_ CC /\ 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 7265	. . . 4 |- ((A (_ CC /\ B (_ 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 799	. . 3 |- ((A (_ CC /\ B (_ CC /\ 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 389	. 2 |- (((A (_ CC /\ B (_ CC /\ 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 7265	. . . 4 |- ((A (_ CC /\ 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 798	. . 3 |- ((A (_ CC /\ B (_ CC /\ 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 389	. 2 |- (((A (_ CC /\ B (_ CC /\ 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 543	1 |- (((A (_ CC /\ B (_ CC /\ 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 146   /\ wa 223   /\ w3a 775   e. wcel 958  A.wral 1645  E.wrex 1646   (_ wss 2047   class class class wbr 2619  ran crn 3171   Fn wfn 3177  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232   - cmin 5292  RR+crp 5300   < clt 5486  abscabs 6750  -cn->ccncf 7262

This theorem is referenced by:  isupivth 7290

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-qs 4266  df-ni 5000  df-nq 5038  df-np 5086  df-nr 5167  df-c 5240  df-cncf 7263

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