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Metamath Proof Explorer

Theorem elcncf1ii

Description: Membership in the set of continuous complex functions from A to B . (Contributed by Paul Chapman, 26-Nov-2007)

Ref Expression
Hypotheses elcncf1i.1 
|- F : A --> B
elcncf1i.2 
|- ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ )
elcncf1i.3 
|- ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
Assertion elcncf1ii 
|- ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) )

Proof

Step Hyp Ref Expression
1 elcncf1i.1 
 |-  F : A --> B
2 elcncf1i.2 
 |-  ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ )
3 elcncf1i.3 
 |-  ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
4 1 a1i 
 |-  ( T. -> F : A --> B )
5 2 a1i 
 |-  ( T. -> ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ ) )
6 3 a1i 
 |-  ( T. -> ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )
7 4 5 6 elcncf1di 
 |-  ( T. -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )
8 7 mptru 
 |-  ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) )