Theorem lanval2

Description: The set of left Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by isup2 and upciclem1 . (Contributed by Zhi Wang, 3-Nov-2025)

	Ref	Expression
Hypotheses	islan.r	|- R = ( D FuncCat E )
	islan.s	|- S = ( C FuncCat E )
	islan.k	|- K = ( <. D , E >. -o.F F )
Assertion	lanval2	|- ( F e. ( C Func D ) -> ( F ( <. C , D >. Lan E ) X ) = ( K ( R UP S ) X ) )

Proof

Step	Hyp	Ref	Expression
1		islan.r	|- R = ( D FuncCat E )
2		islan.s	|- S = ( C FuncCat E )
3		islan.k	|- K = ( <. D , E >. -o.F F )
4	1 2 3	islan	|- ( x e. ( F ( <. C , D >. Lan E ) X ) -> x e. ( K ( R UP S ) X ) )
5	4	adantl	|- ( ( F e. ( C Func D ) /\ x e. ( F ( <. C , D >. Lan E ) X ) ) -> x e. ( K ( R UP S ) X ) )
6		simpr	|- ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> x e. ( K ( R UP S ) X ) )
7		simpl	|- ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> F e. ( C Func D ) )
8	2	fucbas	|- ( C Func E ) = ( Base ` S )
9	8	uprcl	|- ( x e. ( K ( R UP S ) X ) -> ( K e. ( R Func S ) /\ X e. ( C Func E ) ) )
10	9	simprd	|- ( x e. ( K ( R UP S ) X ) -> X e. ( C Func E ) )
11	10	adantl	|- ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> X e. ( C Func E ) )
12	3	eqcomi	|- ( <. D , E >. -o.F F ) = K
13	12	a1i	|- ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> ( <. D , E >. -o.F F ) = K )
14	1 2 7 11 13	lanval	|- ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> ( F ( <. C , D >. Lan E ) X ) = ( K ( R UP S ) X ) )
15	6 14	eleqtrrd	|- ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> x e. ( F ( <. C , D >. Lan E ) X ) )
16	5 15	impbida	|- ( F e. ( C Func D ) -> ( x e. ( F ( <. C , D >. Lan E ) X ) <-> x e. ( K ( R UP S ) X ) ) )
17	16	eqrdv	|- ( F e. ( C Func D ) -> ( F ( <. C , D >. Lan E ) X ) = ( K ( R UP S ) X ) )