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Theorem elcncf 8527
Description: Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 11-Oct-2007.)
Assertion
Ref Expression
elcncf |- ((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))))
Distinct variable groups:   w,A,x,y,z   w,F,x,y,z
Proof of Theorem elcncf
StepHypRef Expression
1 cncfval 8526 . . . . 5 |- ((A C_ CC /\ B C_ CC) -> (A-cn->B) = {f | (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))})
21eleq2d 1964 . . . 4 |- ((A C_ CC /\ B C_ CC) -> (F e. (A-cn->B) <-> F e. {f | (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))}))
3 feq1 4551 . . . . . 6 |- (f = F -> (f:A-->B <-> F:A-->B))
4 fveq1 4680 . . . . . . . . . . . 12 |- (f = F -> (f` x) = (F` x))
5 fveq1 4680 . . . . . . . . . . . 12 |- (f = F -> (f` w) = (F` w))
64, 5opreq12d 4900 . . . . . . . . . . 11 |- (f = F -> ((f` x) - (f` w)) = ((F` x) - (F` w)))
76fveq2d 4685 . . . . . . . . . 10 |- (f = F -> (abs` ((f` x) - (f` w))) = (abs` ((F` x) - (F` w))))
87breq1d 3348 . . . . . . . . 9 |- (f = F -> ((abs` ((f` x) - (f` w))) < y <-> (abs` ((F` x) - (F` w))) < y))
98imbi2d 674 . . . . . . . 8 |- (f = F -> (((abs` (x - w)) < z -> (abs` ((f` x) - (f` w))) < y) <-> ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)))
109rexralbidv 2142 . . . . . . 7 |- (f = F -> (E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((f` x) - (f` w))) < y) <-> E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)))
11102ralbidv 2140 . . . . . 6 |- (f = F -> (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) <-> 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)))
123, 11anbi12d 690 . . . . 5 |- (f = F -> ((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-->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))))
1312elabg 2405 . . . 4 |- (F e. _V -> (F e. {f | (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-->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))))
142, 13sylan9bb 599 . . 3 |- (((A C_ CC /\ B C_ CC) /\ F e. _V) -> (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))))
1514ex 402 . 2 |- ((A C_ CC /\ B C_ CC) -> (F e. _V -> (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)))))
16 elisset 2299 . . . . . . 7 |- (F e. (A-cn->B) -> F e. _V)
1716adantl 424 . . . . . 6 |- (((A C_ CC /\ B C_ CC) /\ F e. (A-cn->B)) -> F e. _V)
18 axcnex 6419 . . . . . . . . . 10 |- CC e. _V
1918ssex 3455 . . . . . . . . 9 |- (A C_ CC -> A e. _V)
20 fex 4595 . . . . . . . . . . 11 |- ((F:A-->B /\ A e. _V) -> F e. _V)
2120adantlr 429 . . . . . . . . . 10 |- (((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)) /\ A e. _V) -> F e. _V)
2221expcom 403 . . . . . . . . 9 |- (A e. _V -> ((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 e. _V))
2319, 22syl 12 . . . . . . . 8 |- (A C_ CC -> ((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 e. _V))
2423adantr 425 . . . . . . 7 |- ((A C_ CC /\ B C_ CC) -> ((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 e. _V))
2524imp 377 . . . . . 6 |- (((A C_ CC /\ B C_ CC) /\ (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 e. _V)
2617, 25jaodan 471 . . . . 5 |- (((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)))) -> F e. _V)
2726ex 402 . . . 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))) -> F e. _V))
2827con3d 111 . . 3 |- ((A C_ CC /\ B C_ CC) -> (-. F e. _V -> -. (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)))))
29 ioran 331 . . . 4 |- (-. (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))) <-> (-. 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))))
30 pm5.21 741 . . . 4 |- ((-. 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))) -> (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))))
3129, 30sylbi 216 . . 3 |- (-. (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))) -> (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))))
3228, 31syl6 25 . 2 |- ((A C_ CC /\ B C_ CC) -> (-. F e. _V -> (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)))))
3315, 32pm2.61d 141 1 |- ((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))))
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593   class class class wbr 3338  -->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:  cncff 8528  cncffvelrnOLD 8529  negfcncfi 8531  elcncf1di 8532  rescncf 8534  cncffvrn 8535  ivthlem8 8550  efcn 8688  cncfmet 9183
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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