Theorem cofu1st2nd

Description: Rewrite the functor composition with separated functor parts. (Contributed by Zhi Wang, 15-Nov-2025)

	Ref	Expression
Hypotheses	cofu1st2nd.f	|- ( ph -> F e. ( C Func D ) )
	cofu1st2nd.g	|- ( ph -> G e. ( D Func E ) )
Assertion	cofu1st2nd	|- ( ph -> ( G o.func F ) = ( <. ( 1st ` G ) , ( 2nd ` G ) >. o.func <. ( 1st ` F ) , ( 2nd ` F ) >. ) )

Proof

Step	Hyp	Ref	Expression
1		cofu1st2nd.f	|- ( ph -> F e. ( C Func D ) )
2		cofu1st2nd.g	|- ( ph -> G e. ( D Func E ) )
3		relfunc	|- Rel ( D Func E )
4		1st2nd	|- ( ( Rel ( D Func E ) /\ G e. ( D Func E ) ) -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
5	3 2 4	sylancr	|- ( ph -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
6		relfunc	|- Rel ( C Func D )
7		1st2nd	|- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
8	6 1 7	sylancr	|- ( ph -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
9	5 8	oveq12d	|- ( ph -> ( G o.func F ) = ( <. ( 1st ` G ) , ( 2nd ` G ) >. o.func <. ( 1st ` F ) , ( 2nd ` F ) >. ) )