Theorem ranval2

Description: The set of right Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by oppcup2 and oppcup3lem . (Contributed by Zhi Wang, 4-Nov-2025)

	Ref	Expression
Hypotheses	isran.o	$$⊢O=\mathrm{oppCat}\left(D\mathrm{FuncCat}E\right)$$
	isran.p	$$⊢P=\mathrm{oppCat}\left(C\mathrm{FuncCat}E\right)$$
	isran.k	No typesetting found for |- ( ph -> ( <. D , E >. -o.F F ) = <. J , K >. ) with typecode |-
	ranval2.f	$$⊢\phi \to F\in \left(C\mathrm{Func}D\right)$$
Assertion	ranval2	Could not format assertion : No typesetting found for |- ( ph -> ( F ( <. C , D >. Ran E ) X ) = ( <. J , tpos K >. ( O UP P ) X ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		isran.o	$$⊢O=\mathrm{oppCat}\left(D\mathrm{FuncCat}E\right)$$
2		isran.p	$$⊢P=\mathrm{oppCat}\left(C\mathrm{FuncCat}E\right)$$
3		isran.k	Could not format ( ph -> ( <. D , E >. -o.F F ) = <. J , K >. ) : No typesetting found for |- ( ph -> ( <. D , E >. -o.F F ) = <. J , K >. ) with typecode |-
4		ranval2.f	$$⊢\phi \to F\in \left(C\mathrm{Func}D\right)$$
5	3	adantr	Could not format ( ( ph /\ x e. ( F ( <. C , D >. Ran E ) X ) ) -> ( <. D , E >. -o.F F ) = <. J , K >. ) : No typesetting found for |- ( ( ph /\ x e. ( F ( <. C , D >. Ran E ) X ) ) -> ( <. D , E >. -o.F F ) = <. J , K >. ) with typecode |-
6		simpr	Could not format ( ( ph /\ x e. ( F ( <. C , D >. Ran E ) X ) ) -> x e. ( F ( <. C , D >. Ran E ) X ) ) : No typesetting found for |- ( ( ph /\ x e. ( F ( <. C , D >. Ran E ) X ) ) -> x e. ( F ( <. C , D >. Ran E ) X ) ) with typecode |-
7	1 2 5 6	isran	Could not format ( ( ph /\ x e. ( F ( <. C , D >. Ran E ) X ) ) -> x e. ( <. J , tpos K >. ( O UP P ) X ) ) : No typesetting found for |- ( ( ph /\ x e. ( F ( <. C , D >. Ran E ) X ) ) -> x e. ( <. J , tpos K >. ( O UP P ) X ) ) with typecode |-
8		simpr	Could not format ( ( ph /\ x e. ( <. J , tpos K >. ( O UP P ) X ) ) -> x e. ( <. J , tpos K >. ( O UP P ) X ) ) : No typesetting found for |- ( ( ph /\ x e. ( <. J , tpos K >. ( O UP P ) X ) ) -> x e. ( <. J , tpos K >. ( O UP P ) X ) ) with typecode |-
9		eqid	$$⊢D\mathrm{FuncCat}E=D\mathrm{FuncCat}E$$
10		eqid	$$⊢C\mathrm{FuncCat}E=C\mathrm{FuncCat}E$$
11	4	adantr	Could not format ( ( ph /\ x e. ( <. J , tpos K >. ( O UP P ) X ) ) -> F e. ( C Func D ) ) : No typesetting found for |- ( ( ph /\ x e. ( <. J , tpos K >. ( O UP P ) X ) ) -> F e. ( C Func D ) ) with typecode |-
12	10	fucbas	$$⊢C\mathrm{Func}E={\mathrm{Base}}_{\left(C\mathrm{FuncCat}E\right)}$$
13	2 12	oppcbas	$$⊢C\mathrm{Func}E={\mathrm{Base}}_P$$
14	13	uprcl	Could not format ( x e. ( <. J , tpos K >. ( O UP P ) X ) -> ( <. J , tpos K >. e. ( O Func P ) /\ X e. ( C Func E ) ) ) : No typesetting found for |- ( x e. ( <. J , tpos K >. ( O UP P ) X ) -> ( <. J , tpos K >. e. ( O Func P ) /\ X e. ( C Func E ) ) ) with typecode |-
15	14	simprd	Could not format ( x e. ( <. J , tpos K >. ( O UP P ) X ) -> X e. ( C Func E ) ) : No typesetting found for |- ( x e. ( <. J , tpos K >. ( O UP P ) X ) -> X e. ( C Func E ) ) with typecode |-
16	15	adantl	Could not format ( ( ph /\ x e. ( <. J , tpos K >. ( O UP P ) X ) ) -> X e. ( C Func E ) ) : No typesetting found for |- ( ( ph /\ x e. ( <. J , tpos K >. ( O UP P ) X ) ) -> X e. ( C Func E ) ) with typecode |-
17	3	adantr	Could not format ( ( ph /\ x e. ( <. J , tpos K >. ( O UP P ) X ) ) -> ( <. D , E >. -o.F F ) = <. J , K >. ) : No typesetting found for |- ( ( ph /\ x e. ( <. J , tpos K >. ( O UP P ) X ) ) -> ( <. D , E >. -o.F F ) = <. J , K >. ) with typecode |-
18	9 10 11 16 17 1 2	ranval	Could not format ( ( ph /\ x e. ( <. J , tpos K >. ( O UP P ) X ) ) -> ( F ( <. C , D >. Ran E ) X ) = ( <. J , tpos K >. ( O UP P ) X ) ) : No typesetting found for |- ( ( ph /\ x e. ( <. J , tpos K >. ( O UP P ) X ) ) -> ( F ( <. C , D >. Ran E ) X ) = ( <. J , tpos K >. ( O UP P ) X ) ) with typecode |-
19	8 18	eleqtrrd	Could not format ( ( ph /\ x e. ( <. J , tpos K >. ( O UP P ) X ) ) -> x e. ( F ( <. C , D >. Ran E ) X ) ) : No typesetting found for |- ( ( ph /\ x e. ( <. J , tpos K >. ( O UP P ) X ) ) -> x e. ( F ( <. C , D >. Ran E ) X ) ) with typecode |-
20	7 19	impbida	Could not format ( ph -> ( x e. ( F ( <. C , D >. Ran E ) X ) <-> x e. ( <. J , tpos K >. ( O UP P ) X ) ) ) : No typesetting found for |- ( ph -> ( x e. ( F ( <. C , D >. Ran E ) X ) <-> x e. ( <. J , tpos K >. ( O UP P ) X ) ) ) with typecode |-
21	20	eqrdv	Could not format ( ph -> ( F ( <. C , D >. Ran E ) X ) = ( <. J , tpos K >. ( O UP P ) X ) ) : No typesetting found for |- ( ph -> ( F ( <. C , D >. Ran E ) X ) = ( <. J , tpos K >. ( O UP P ) X ) ) with typecode |-