Theorem mpoeq3dv

Description: An equality deduction for the maps-to notation restricted to the value of the operation. (Contributed by SO, 16-Jul-2018)

	Ref	Expression
Hypothesis	mpoeq3dv.1	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	mpoeq3dv	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		mpoeq3dv.1	⊢ ( 𝜑 → 𝐶 = 𝐷 )
2	1	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝐶 = 𝐷 )
3	2	mpoeq3dva	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐷 ) )