Theorem islan

Description: A left Kan extension is a universal pair. (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	islan	Could not format assertion : No typesetting found for |- ( L e. ( F ( <. C , D >. Lan E ) X ) -> L e. ( 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		id	Could not format ( L e. ( F ( <. C , D >. Lan E ) X ) -> L e. ( F ( <. C , D >. Lan E ) X ) ) : No typesetting found for |- ( L e. ( F ( <. C , D >. Lan E ) X ) -> L e. ( F ( <. C , D >. Lan E ) X ) ) with typecode |-
5		lanrcl	Could not format ( L e. ( F ( <. C , D >. Lan E ) X ) -> ( F e. ( C Func D ) /\ X e. ( C Func E ) ) ) : No typesetting found for |- ( L e. ( F ( <. C , D >. Lan E ) X ) -> ( F e. ( C Func D ) /\ X e. ( C Func E ) ) ) with typecode |-
6	5	simpld	Could not format ( L e. ( F ( <. C , D >. Lan E ) X ) -> F e. ( C Func D ) ) : No typesetting found for |- ( L e. ( F ( <. C , D >. Lan E ) X ) -> F e. ( C Func D ) ) with typecode |-
7	5	simprd	Could not format ( L e. ( F ( <. C , D >. Lan E ) X ) -> X e. ( C Func E ) ) : No typesetting found for |- ( L e. ( F ( <. C , D >. Lan E ) X ) -> X e. ( C Func E ) ) with typecode |-
8	3	eqcomi	Could not format ( <. D , E >. -o.F F ) = K : No typesetting found for |- ( <. D , E >. -o.F F ) = K with typecode |-
9	8	a1i	Could not format ( L e. ( F ( <. C , D >. Lan E ) X ) -> ( <. D , E >. -o.F F ) = K ) : No typesetting found for |- ( L e. ( F ( <. C , D >. Lan E ) X ) -> ( <. D , E >. -o.F F ) = K ) with typecode |-
10	1 2 6 7 9	lanval	Could not format ( L e. ( F ( <. C , D >. Lan E ) X ) -> ( F ( <. C , D >. Lan E ) X ) = ( K ( R UP S ) X ) ) : No typesetting found for |- ( L e. ( F ( <. C , D >. Lan E ) X ) -> ( F ( <. C , D >. Lan E ) X ) = ( K ( R UP S ) X ) ) with typecode |-
11	4 10	eleqtrd	Could not format ( L e. ( F ( <. C , D >. Lan E ) X ) -> L e. ( K ( R UP S ) X ) ) : No typesetting found for |- ( L e. ( F ( <. C , D >. Lan E ) X ) -> L e. ( K ( R UP S ) X ) ) with typecode |-