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Metamath Proof Explorer

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 )