Theorem offun

Description: The function operation produces a function. (Contributed by SN, 23-Jul-2024)

	Ref	Expression
Hypotheses	offun.1	|- ( ph -> F Fn A )
	offun.2	|- ( ph -> G Fn B )
	offun.3	|- ( ph -> A e. V )
	offun.4	|- ( ph -> B e. W )
Assertion	offun	|- ( ph -> Fun ( F oF R G ) )

Proof

Step	Hyp	Ref	Expression
1		offun.1	|- ( ph -> F Fn A )
2		offun.2	|- ( ph -> G Fn B )
3		offun.3	|- ( ph -> A e. V )
4		offun.4	|- ( ph -> B e. W )
5		eqid	|- ( A i^i B ) = ( A i^i B )
6	1 2 3 4 5	offn	|- ( ph -> ( F oF R G ) Fn ( A i^i B ) )
7		fnfun	|- ( ( F oF R G ) Fn ( A i^i B ) -> Fun ( F oF R G ) )
8	6 7	syl	|- ( ph -> Fun ( F oF R G ) )