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Theorem elcncf 7200
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 (_ 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))))
Distinct variable groups:   w,A,x,y,z   w,F,x,y,z
Proof of Theorem elcncf
StepHypRef Expression
1 cncfval 7199 . . . . 5 |- ((A (_ CC /\ B (_ 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 1533 . . . 4 |- ((A (_ CC /\ B (_ 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 3606 . . . . . 6 |- (f = F -> (f:A-->B <-> F:A-->B))
4 fveq1 3708 . . . . . . . . . . . 12 |- (f = F -> (f` x) = (F` x))
5 fveq1 3708 . . . . . . . . . . . 12 |- (f = F -> (f` w) = (F` w))
64, 5opreq12d 3963 . . . . . . . . . . 11 |- (f = F -> ((f` x) - (f` w)) = ((F` x) - (F` w)))
76fveq2d 3713 . . . . . . . . . 10 |- (f = F -> (abs` ((f` x) - (f` w))) = (abs` ((F` x) - (F` w))))
87breq1d 2619 . . . . . . . . 9 |- (f = F -> ((abs` ((f` x) - (f` w))) < y <-> (abs` ((F` x) - (F` w))) < y))
98imbi2d 610 . . . . . . . 8 |- (f = F -> (((abs` (x - w)) < z -> (abs` ((f` x) - (f` w))) < y) <-> ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)))
109rexralbidv 1674 . . . . . . 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 1672 . . . . . 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 626 . . . . 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 1890 . . . 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 538 . . 3 |- (((A (_ CC /\ B (_ 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 373 . 2 |- ((A (_ CC /\ B (_ 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 1808 . . . . . . 7 |- (F e. (A-cn->B) -> F e. V)
1716adantl 388 . . . . . 6 |- (((A (_ CC /\ B (_ CC) /\ F e. (A-cn->B)) -> F e. V)
18 axcnex 5239 . . . . . . . . . 10 |- CC e. V
1918ssex 2709 . . . . . . . . 9 |- (A (_ CC -> A e. V)
20 fex 3637 . . . . . . . . . . 11 |- ((F:A-->B /\ A e. V) -> F e. V)
2120adantlr 393 . . . . . . . . . 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 374 . . . . . . . . 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 10 . . . . . . . 8 |- (A (_ 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 389 . . . . . . 7 |- ((A (_ CC /\ B (_ 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 350 . . . . . 6 |- (((A (_ CC /\ B (_ 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 426 . . . . 5 |- (((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)))) -> F e. V)
2726ex 373 . . . 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))) -> F e. V))
2827con3d 95 . . 3 |- ((A (_ CC /\ B (_ 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 306 . . . 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 pm5.21 675 . . . 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 199 . . 3