Theorem oppcuprcl4

Description: Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025)

	Ref	Expression
Hypotheses	oppcuprcl2.x	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) 𝑀 )
	oppcuprcl4.o	⊢ 𝑂 = ( oppCat ‘ 𝐷 )
	oppcuprcl4.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
Assertion	oppcuprcl4	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		oppcuprcl2.x	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) 𝑀 )
2		oppcuprcl4.o	⊢ 𝑂 = ( oppCat ‘ 𝐷 )
3		oppcuprcl4.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
4	2 3	oppcbas	⊢ 𝐵 = ( Base ‘ 𝑂 )
5	1 4	uprcl4	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )