2oppffunc

Description: The opposite functor of an opposite functor is a functor on the original categories. (Contributed by Zhi Wang, 14-Nov-2025) The functor in opposite categories does not have to be an opposite functor. (Revised by Zhi Wang, 17-Nov-2025)

	Ref	Expression
Hypotheses	funcoppc2.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
	funcoppc2.p	⊢ 𝑃 = ( oppCat ‘ 𝐷 )
	funcoppc2.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
	funcoppc2.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑊 )
	2oppffunc.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑂 Func 𝑃 ) )
Assertion	2oppffunc	⊢ ( 𝜑 → ( oppFunc ‘ 𝐹 ) ∈ ( 𝐶 Func 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		funcoppc2.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
2		funcoppc2.p	⊢ 𝑃 = ( oppCat ‘ 𝐷 )
3		funcoppc2.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
4		funcoppc2.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑊 )
5		2oppffunc.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑂 Func 𝑃 ) )
6		oppfval2	⊢ ( 𝐹 ∈ ( 𝑂 Func 𝑃 ) → ( oppFunc ‘ 𝐹 ) = ⟨ ( 1^st ‘ 𝐹 ) , tpos ( 2^nd ‘ 𝐹 ) ⟩ )
7	5 6	syl	⊢ ( 𝜑 → ( oppFunc ‘ 𝐹 ) = ⟨ ( 1^st ‘ 𝐹 ) , tpos ( 2^nd ‘ 𝐹 ) ⟩ )
8	5	func1st2nd	⊢ ( 𝜑 → ( 1^st ‘ 𝐹 ) ( 𝑂 Func 𝑃 ) ( 2^nd ‘ 𝐹 ) )
9	1 2 3 4 8	funcoppc2	⊢ ( 𝜑 → ( 1^st ‘ 𝐹 ) ( 𝐶 Func 𝐷 ) tpos ( 2^nd ‘ 𝐹 ) )
10		df-br	⊢ ( ( 1^st ‘ 𝐹 ) ( 𝐶 Func 𝐷 ) tpos ( 2^nd ‘ 𝐹 ) ↔ ⟨ ( 1^st ‘ 𝐹 ) , tpos ( 2^nd ‘ 𝐹 ) ⟩ ∈ ( 𝐶 Func 𝐷 ) )
11	9 10	sylib	⊢ ( 𝜑 → ⟨ ( 1^st ‘ 𝐹 ) , tpos ( 2^nd ‘ 𝐹 ) ⟩ ∈ ( 𝐶 Func 𝐷 ) )
12	7 11	eqeltrd	⊢ ( 𝜑 → ( oppFunc ‘ 𝐹 ) ∈ ( 𝐶 Func 𝐷 ) )

Metamath Proof Explorer

Theorem 2oppffunc

Proof