Theorem lanrcl3

Description: Reverse closure for left Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025)

	Ref	Expression
Hypothesis	lanrcl2.l	No typesetting found for |- ( ph -> L ( F ( <. C , D >. Lan E ) X ) A ) with typecode |-
Assertion	lanrcl3	$$⊢\phi \to X\in \left(C\mathrm{Func}E\right)$$

Proof

Step	Hyp	Ref	Expression
1		lanrcl2.l	Could not format ( ph -> L ( F ( <. C , D >. Lan E ) X ) A ) : No typesetting found for |- ( ph -> L ( F ( <. C , D >. Lan E ) X ) A ) with typecode |-
2		df-br	Could not format ( L ( F ( <. C , D >. Lan E ) X ) A <-> <. L , A >. e. ( F ( <. C , D >. Lan E ) X ) ) : No typesetting found for |- ( L ( F ( <. C , D >. Lan E ) X ) A <-> <. L , A >. e. ( F ( <. C , D >. Lan E ) X ) ) with typecode |-
3	1 2	sylib	Could not format ( ph -> <. L , A >. e. ( F ( <. C , D >. Lan E ) X ) ) : No typesetting found for |- ( ph -> <. L , A >. e. ( F ( <. C , D >. Lan E ) X ) ) with typecode |-
4		lanrcl	Could not format ( <. L , A >. e. ( F ( <. C , D >. Lan E ) X ) -> ( F e. ( C Func D ) /\ X e. ( C Func E ) ) ) : No typesetting found for |- ( <. L , A >. e. ( F ( <. C , D >. Lan E ) X ) -> ( F e. ( C Func D ) /\ X e. ( C Func E ) ) ) with typecode |-
5	3 4	syl	$$⊢\phi \to \left(F\in \left(C\mathrm{Func}D\right)\wedge X\in \left(C\mathrm{Func}E\right)\right)$$
6	5	simprd	$$⊢\phi \to X\in \left(C\mathrm{Func}E\right)$$