Theorem prstcle

Description: Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024)

	Ref	Expression
Hypotheses	prstcnid.c	|- ( ph -> C = ( ProsetToCat ` K ) )
	prstcnid.k	|- ( ph -> K e. Proset )
	prstcle.l	|- ( ph -> .<_ = ( le ` K ) )
Assertion	prstcle	|- ( ph -> ( X .<_ Y <-> X ( le ` C ) Y ) )

Proof

Step	Hyp	Ref	Expression
1		prstcnid.c	|- ( ph -> C = ( ProsetToCat ` K ) )
2		prstcnid.k	|- ( ph -> K e. Proset )
3		prstcle.l	|- ( ph -> .<_ = ( le ` K ) )
4	1 2 3	prstcleval	|- ( ph -> .<_ = ( le ` C ) )
5	4	breqd	|- ( ph -> ( X .<_ Y <-> X ( le ` C ) Y ) )