Theorem opf2fval

Description: The morphism part of the op functor on functor categories. Lemma for fucoppc . (Contributed by Zhi Wang, 18-Nov-2025)

	Ref	Expression
Hypotheses	opf2fval.f	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑦 𝑁 𝑥 ) ) ) )
	opf2fval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
	opf2fval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	opf2fval	⊢ ( 𝜑 → ( 𝑋 𝐹 𝑌 ) = ( I ↾ ( 𝑌 𝑁 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		opf2fval.f	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑦 𝑁 𝑥 ) ) ) )
2		opf2fval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
3		opf2fval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
4		simprr	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝑦 = 𝑌 )
5		simprl	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝑥 = 𝑋 )
6	4 5	oveq12d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑦 𝑁 𝑥 ) = ( 𝑌 𝑁 𝑋 ) )
7	6	reseq2d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( I ↾ ( 𝑦 𝑁 𝑥 ) ) = ( I ↾ ( 𝑌 𝑁 𝑋 ) ) )
8		ovexd	⊢ ( 𝜑 → ( 𝑌 𝑁 𝑋 ) ∈ V )
9		resiexg	⊢ ( ( 𝑌 𝑁 𝑋 ) ∈ V → ( I ↾ ( 𝑌 𝑁 𝑋 ) ) ∈ V )
10	8 9	syl	⊢ ( 𝜑 → ( I ↾ ( 𝑌 𝑁 𝑋 ) ) ∈ V )
11	1 7 2 3 10	ovmpod	⊢ ( 𝜑 → ( 𝑋 𝐹 𝑌 ) = ( I ↾ ( 𝑌 𝑁 𝑋 ) ) )