Theorem prcof21a

Description: The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025)

	Ref	Expression
Hypotheses	prcof21a.n	|- N = ( D Nat E )
	prcof21a.a	|- ( ph -> A e. ( K N L ) )
	prcof21a.p	|- ( ph -> ( 2nd ` ( <. D , E >. -o.F F ) ) = P )
	prcof21a.f	|- ( ph -> F e. U )
Assertion	prcof21a	|- ( ph -> ( ( K P L ) ` A ) = ( A o. ( 1st ` F ) ) )

Proof

Step	Hyp	Ref	Expression
1		prcof21a.n	|- N = ( D Nat E )
2		prcof21a.a	|- ( ph -> A e. ( K N L ) )
3		prcof21a.p	|- ( ph -> ( 2nd ` ( <. D , E >. -o.F F ) ) = P )
4		prcof21a.f	|- ( ph -> F e. U )
5	1	natrcl	|- ( A e. ( K N L ) -> ( K e. ( D Func E ) /\ L e. ( D Func E ) ) )
6	2 5	syl	|- ( ph -> ( K e. ( D Func E ) /\ L e. ( D Func E ) ) )
7	6	simpld	|- ( ph -> K e. ( D Func E ) )
8	6	simprd	|- ( ph -> L e. ( D Func E ) )
9	1 7 8 3 4	prcof2a	|- ( ph -> ( K P L ) = ( a e. ( K N L ) |-> ( a o. ( 1st ` F ) ) ) )
10		simpr	|- ( ( ph /\ a = A ) -> a = A )
11	10	coeq1d	|- ( ( ph /\ a = A ) -> ( a o. ( 1st ` F ) ) = ( A o. ( 1st ` F ) ) )
12		fvexd	|- ( ph -> ( 1st ` F ) e. _V )
13	2 12	coexd	|- ( ph -> ( A o. ( 1st ` F ) ) e. _V )
14	9 11 2 13	fvmptd	|- ( ph -> ( ( K P L ) ` A ) = ( A o. ( 1st ` F ) ) )