Theorem ovmpt4d

Description: Deduction version of ovmpt4g . (This is the operation analogue of fvmpt2d .) (Contributed by Zhi Wang, 9-Oct-2025)

	Ref	Expression
Hypotheses	ovmpt4d.1	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) )
	ovmpt4d.2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐶 ∈ 𝑉 )
Assertion	ovmpt4d	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 𝐹 𝑦 ) = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ovmpt4d.1	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) )
2		ovmpt4d.2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐶 ∈ 𝑉 )
3	1	oveqdr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑦 ) )
4		simprl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑥 ∈ 𝐴 )
5		simprr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐵 )
6		eqid	⊢ ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
7	6	ovmpt4g	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ) → ( 𝑥 ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑦 ) = 𝐶 )
8	4 5 2 7	syl3anc	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑦 ) = 𝐶 )
9	3 8	eqtrd	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 𝐹 𝑦 ) = 𝐶 )