Theorem oppcuprcl5

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 )
	oppcuprcl5.j	⊢ 𝐽 = ( Hom ‘ 𝐸 )
Assertion	oppcuprcl5	⊢ ( 𝜑 → 𝑀 ∈ ( ( 𝐹 ‘ 𝑋 ) 𝐽 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		oppcuprcl2.x	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) 𝑀 )
2		oppcuprcl2.p	⊢ 𝑃 = ( oppCat ‘ 𝐸 )
3		oppcuprcl5.j	⊢ 𝐽 = ( Hom ‘ 𝐸 )
4		eqid	⊢ ( Hom ‘ 𝑃 ) = ( Hom ‘ 𝑃 )
5	1 4	uprcl5	⊢ ( 𝜑 → 𝑀 ∈ ( 𝑊 ( Hom ‘ 𝑃 ) ( 𝐹 ‘ 𝑋 ) ) )
6	3 2	oppchom	⊢ ( 𝑊 ( Hom ‘ 𝑃 ) ( 𝐹 ‘ 𝑋 ) ) = ( ( 𝐹 ‘ 𝑋 ) 𝐽 𝑊 )
7	5 6	eleqtrdi	⊢ ( 𝜑 → 𝑀 ∈ ( ( 𝐹 ‘ 𝑋 ) 𝐽 𝑊 ) )