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	⊢ ( 𝜑 → 𝐶 = ( ProsetToCat ‘ 𝐾 ) )
	prstcnid.k	⊢ ( 𝜑 → 𝐾 ∈ Proset )
	prstcle.l	⊢ ( 𝜑 → ≤ = ( le ‘ 𝐾 ) )
Assertion	prstcle	⊢ ( 𝜑 → ( 𝑋 ≤ 𝑌 ↔ 𝑋 ( le ‘ 𝐶 ) 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		prstcnid.c	⊢ ( 𝜑 → 𝐶 = ( ProsetToCat ‘ 𝐾 ) )
2		prstcnid.k	⊢ ( 𝜑 → 𝐾 ∈ Proset )
3		prstcle.l	⊢ ( 𝜑 → ≤ = ( le ‘ 𝐾 ) )
4	1 2 3	prstcleval	⊢ ( 𝜑 → ≤ = ( le ‘ 𝐶 ) )
5	4	breqd	⊢ ( 𝜑 → ( 𝑋 ≤ 𝑌 ↔ 𝑋 ( le ‘ 𝐶 ) 𝑌 ) )