Theorem dmmpoga

Description: Domain of an operation given by the maps-to notation, closed form of dmmpo . (Contributed by Alexander van der Vekens, 10-Feb-2019) (Proof shortened by Lammen, 29-May-2024)

	Ref	Expression
Hypothesis	dmmpog.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
Assertion	dmmpoga	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = ( 𝐴 × 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		dmmpog.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
2	1	fnmpo	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → 𝐹 Fn ( 𝐴 × 𝐵 ) )
3	2	fndmd	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = ( 𝐴 × 𝐵 ) )