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Metamath Proof Explorer

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 ) )