Theorem fnfco

Description: Composition of two functions. (Contributed by NM, 22-May-2006)

	Ref	Expression
Assertion	fnfco	|- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B )

Proof

Step	Hyp	Ref	Expression
1		df-f	|- ( G : B --> A <-> ( G Fn B /\ ran G C_ A ) )
2		fnco	|- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B )
3	2	3expb	|- ( ( F Fn A /\ ( G Fn B /\ ran G C_ A ) ) -> ( F o. G ) Fn B )
4	1 3	sylan2b	|- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B )