Theorem funcfn2

Description: The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017)

	Ref	Expression
Hypotheses	funcfn2.b	|- B = ( Base ` D )
	funcfn2.f	|- ( ph -> F ( D Func E ) G )
Assertion	funcfn2	|- ( ph -> G Fn ( B X. B ) )

Proof

Step	Hyp	Ref	Expression
1		funcfn2.b	|- B = ( Base ` D )
2		funcfn2.f	|- ( ph -> F ( D Func E ) G )
3		eqid	|- ( Hom ` D ) = ( Hom ` D )
4		eqid	|- ( Hom ` E ) = ( Hom ` E )
5	1 3 4 2	funcixp	|- ( ph -> G e. X_ x e. ( B X. B ) ( ( ( F ` ( 1st ` x ) ) ( Hom ` E ) ( F ` ( 2nd ` x ) ) ) ^m ( ( Hom ` D ) ` x ) ) )
6		ixpfn	|- ( G e. X_ x e. ( B X. B ) ( ( ( F ` ( 1st ` x ) ) ( Hom ` E ) ( F ` ( 2nd ` x ) ) ) ^m ( ( Hom ` D ) ` x ) ) -> G Fn ( B X. B ) )
7	5 6	syl	|- ( ph -> G Fn ( B X. B ) )