Theorem mbff

Description: A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014)

	Ref	Expression
Assertion	mbff	|- ( F e. MblFn -> F : dom F --> CC )

Proof

Step	Hyp	Ref	Expression
1		ismbf1	|- ( F e. MblFn <-> ( F e. ( CC ^pm RR ) /\ A. x e. ran (,) ( ( `' ( Re o. F ) " x ) e. dom vol /\ ( `' ( Im o. F ) " x ) e. dom vol ) ) )
2	1	simplbi	|- ( F e. MblFn -> F e. ( CC ^pm RR ) )
3		cnex	|- CC e. _V
4		reex	|- RR e. _V
5	3 4	elpm2	|- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
6	5	simplbi	|- ( F e. ( CC ^pm RR ) -> F : dom F --> CC )
7	2 6	syl	|- ( F e. MblFn -> F : dom F --> CC )