Theorem isran

Description: A right Kan extension is a universal pair. (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 |-
	isran.l	No typesetting found for |- ( ph -> L e. ( F ( <. C , D >. Ran E ) X ) ) with typecode |-
Assertion	isran	Could not format assertion : No typesetting found for |- ( ph -> L e. ( <. 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		isran.l	Could not format ( ph -> L e. ( F ( <. C , D >. Ran E ) X ) ) : No typesetting found for |- ( ph -> L e. ( F ( <. C , D >. Ran E ) X ) ) with typecode |-
5		eqid	$$⊢D\mathrm{FuncCat}E=D\mathrm{FuncCat}E$$
6		eqid	$$⊢C\mathrm{FuncCat}E=C\mathrm{FuncCat}E$$
7		ranrcl	Could not format ( L e. ( F ( <. C , D >. Ran E ) X ) -> ( F e. ( C Func D ) /\ X e. ( C Func E ) ) ) : No typesetting found for |- ( L e. ( F ( <. C , D >. Ran E ) X ) -> ( F e. ( C Func D ) /\ X e. ( C Func E ) ) ) with typecode |-
8	4 7	syl	$$⊢\phi \to \left(F\in \left(C\mathrm{Func}D\right)\wedge X\in \left(C\mathrm{Func}E\right)\right)$$
9	8	simpld	$$⊢\phi \to F\in \left(C\mathrm{Func}D\right)$$
10	8	simprd	$$⊢\phi \to X\in \left(C\mathrm{Func}E\right)$$
11	5 6 9 10 3 1 2	ranval	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 |-
12	4 11	eleqtrd	Could not format ( ph -> L e. ( <. J , tpos K >. ( O UP P ) X ) ) : No typesetting found for |- ( ph -> L e. ( <. J , tpos K >. ( O UP P ) X ) ) with typecode |-