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	$$⊢R=D\mathrm{FuncCat}E$$
	islan.s	$$⊢S=C\mathrm{FuncCat}E$$
	islan.k	No typesetting found for |- K = ( <. D , E >. -o.F F ) with typecode |-
Assertion	lanval2	Could not format assertion : No typesetting found for |- ( F e. ( C Func D ) -> ( F ( <. C , D >. Lan E ) X ) = ( K ( R UP S ) X ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		islan.r	$$⊢R=D\mathrm{FuncCat}E$$
2		islan.s	$$⊢S=C\mathrm{FuncCat}E$$
3		islan.k	Could not format K = ( <. D , E >. -o.F F ) : No typesetting found for |- K = ( <. D , E >. -o.F F ) with typecode |-
4	1 2 3	islan	Could not format ( x e. ( F ( <. C , D >. Lan E ) X ) -> x e. ( K ( R UP S ) X ) ) : No typesetting found for |- ( x e. ( F ( <. C , D >. Lan E ) X ) -> x e. ( K ( R UP S ) X ) ) with typecode |-
5	4	adantl	Could not format ( ( F e. ( C Func D ) /\ x e. ( F ( <. C , D >. Lan E ) X ) ) -> x e. ( K ( R UP S ) X ) ) : No typesetting found for |- ( ( F e. ( C Func D ) /\ x e. ( F ( <. C , D >. Lan E ) X ) ) -> x e. ( K ( R UP S ) X ) ) with typecode |-
6		simpr	Could not format ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> x e. ( K ( R UP S ) X ) ) : No typesetting found for |- ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> x e. ( K ( R UP S ) X ) ) with typecode |-
7		simpl	Could not format ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> F e. ( C Func D ) ) : No typesetting found for |- ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> F e. ( C Func D ) ) with typecode |-
8	2	fucbas	$$⊢C\mathrm{Func}E={\mathrm{Base}}_S$$
9	8	uprcl	Could not format ( x e. ( K ( R UP S ) X ) -> ( K e. ( R Func S ) /\ X e. ( C Func E ) ) ) : No typesetting found for |- ( x e. ( K ( R UP S ) X ) -> ( K e. ( R Func S ) /\ X e. ( C Func E ) ) ) with typecode |-
10	9	simprd	Could not format ( x e. ( K ( R UP S ) X ) -> X e. ( C Func E ) ) : No typesetting found for |- ( x e. ( K ( R UP S ) X ) -> X e. ( C Func E ) ) with typecode |-
11	10	adantl	Could not format ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> X e. ( C Func E ) ) : No typesetting found for |- ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> X e. ( C Func E ) ) with typecode |-
12	3	eqcomi	Could not format ( <. D , E >. -o.F F ) = K : No typesetting found for |- ( <. D , E >. -o.F F ) = K with typecode |-
13	12	a1i	Could not format ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> ( <. D , E >. -o.F F ) = K ) : No typesetting found for |- ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> ( <. D , E >. -o.F F ) = K ) with typecode |-
14	1 2 7 11 13	lanval	Could not format ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> ( F ( <. C , D >. Lan E ) X ) = ( K ( R UP S ) X ) ) : No typesetting found for |- ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> ( F ( <. C , D >. Lan E ) X ) = ( K ( R UP S ) X ) ) with typecode |-
15	6 14	eleqtrrd	Could not format ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> x e. ( F ( <. C , D >. Lan E ) X ) ) : No typesetting found for |- ( ( F e. ( C Func D ) /\ x e. ( K ( R UP S ) X ) ) -> x e. ( F ( <. C , D >. Lan E ) X ) ) with typecode |-
16	5 15	impbida	Could not format ( F e. ( C Func D ) -> ( x e. ( F ( <. C , D >. Lan E ) X ) <-> x e. ( K ( R UP S ) X ) ) ) : No typesetting found for |- ( F e. ( C Func D ) -> ( x e. ( F ( <. C , D >. Lan E ) X ) <-> x e. ( K ( R UP S ) X ) ) ) with typecode |-
17	16	eqrdv	Could not format ( F e. ( C Func D ) -> ( F ( <. C , D >. Lan E ) X ) = ( K ( R UP S ) X ) ) : No typesetting found for |- ( F e. ( C Func D ) -> ( F ( <. C , D >. Lan E ) X ) = ( K ( R UP S ) X ) ) with typecode |-