Theorem funcoppc5

Description: A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 14-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 )
	funcoppc5.f	⊢ ( 𝜑 → ( oppFunc ‘ 𝐹 ) ∈ ( 𝑂 Func 𝑃 ) )
Assertion	funcoppc5	⊢ ( 𝜑 → 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		funcoppc5.f	⊢ ( 𝜑 → ( oppFunc ‘ 𝐹 ) ∈ ( 𝑂 Func 𝑃 ) )
6		relfunc	⊢ Rel ( 𝑂 Func 𝑃 )
7		eqid	⊢ ( oppFunc ‘ 𝐹 ) = ( oppFunc ‘ 𝐹 )
8	5 6 7	oppfrcl	⊢ ( 𝜑 → 𝐹 ∈ ( V × V ) )
9		1st2nd2	⊢ ( 𝐹 ∈ ( V × V ) → 𝐹 = ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ )
10	8 9	syl	⊢ ( 𝜑 → 𝐹 = ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ )
11	10	fveq2d	⊢ ( 𝜑 → ( oppFunc ‘ 𝐹 ) = ( oppFunc ‘ ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ ) )
12		df-ov	⊢ ( ( 1st ‘ 𝐹 ) oppFunc ( 2nd ‘ 𝐹 ) ) = ( oppFunc ‘ ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ )
13	11 12	eqtr4di	⊢ ( 𝜑 → ( oppFunc ‘ 𝐹 ) = ( ( 1st ‘ 𝐹 ) oppFunc ( 2nd ‘ 𝐹 ) ) )
14	13 5	eqeltrrd	⊢ ( 𝜑 → ( ( 1st ‘ 𝐹 ) oppFunc ( 2nd ‘ 𝐹 ) ) ∈ ( 𝑂 Func 𝑃 ) )
15	1 2 3 4 14	funcoppc4	⊢ ( 𝜑 → ( 1st ‘ 𝐹 ) ( 𝐶 Func 𝐷 ) ( 2nd ‘ 𝐹 ) )
16		df-br	⊢ ( ( 1st ‘ 𝐹 ) ( 𝐶 Func 𝐷 ) ( 2nd ‘ 𝐹 ) ↔ ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ ∈ ( 𝐶 Func 𝐷 ) )
17	15 16	sylib	⊢ ( 𝜑 → ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ ∈ ( 𝐶 Func 𝐷 ) )
18	10 17	eqeltrd	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 Func 𝐷 ) )