Theorem oppcuprcl3

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 )
	oppcuprcl2.p	⊢ 𝑃 = ( oppCat ‘ 𝐸 )
	oppcuprcl3.c	⊢ 𝐶 = ( Base ‘ 𝐸 )
Assertion	oppcuprcl3	⊢ ( 𝜑 → 𝑊 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		oppcuprcl2.x	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) 𝑀 )
2		oppcuprcl2.p	⊢ 𝑃 = ( oppCat ‘ 𝐸 )
3		oppcuprcl3.c	⊢ 𝐶 = ( Base ‘ 𝐸 )
4	2 3	oppcbas	⊢ 𝐶 = ( Base ‘ 𝑃 )
5	1 4	uprcl3	⊢ ( 𝜑 → 𝑊 ∈ 𝐶 )