Theorem prstcleval

Description: Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024) (Proof shortened by AV, 12-Nov-2024)

	Ref	Expression
Hypotheses	prstcnid.c	⊢ ( 𝜑 → 𝐶 = ( ProsetToCat ‘ 𝐾 ) )
	prstcnid.k	⊢ ( 𝜑 → 𝐾 ∈ Proset )
	prstcle.l	⊢ ( 𝜑 → ≤ = ( le ‘ 𝐾 ) )
Assertion	prstcleval	⊢ ( 𝜑 → ≤ = ( le ‘ 𝐶 ) )

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		pleid	⊢ le = Slot ( le ‘ ndx )
5		slotsdifplendx2	⊢ ( ( le ‘ ndx ) ≠ ( comp ‘ ndx ) ∧ ( le ‘ ndx ) ≠ ( Hom ‘ ndx ) )
6	5	simpli	⊢ ( le ‘ ndx ) ≠ ( comp ‘ ndx )
7	5	simpri	⊢ ( le ‘ ndx ) ≠ ( Hom ‘ ndx )
8	1 2 4 6 7	prstcnid	⊢ ( 𝜑 → ( le ‘ 𝐾 ) = ( le ‘ 𝐶 ) )
9	3 8	eqtrd	⊢ ( 𝜑 → ≤ = ( le ‘ 𝐶 ) )