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Metamath Proof Explorer

Theorem dmmpog

Description: Domain of an operation given by the maps-to notation, closed form of dmmpo . Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017) (Proof shortened by AV, 10-Feb-2019)

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

Proof

Step	Hyp	Ref	Expression
1		dmmpog.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
2		simpl	⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐶 ∈ 𝑉 )
3	2	ralrimivva	⊢ ( 𝐶 ∈ 𝑉 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 )
4	1	dmmpoga	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = ( 𝐴 × 𝐵 ) )
5	3 4	syl	⊢ ( 𝐶 ∈ 𝑉 → dom 𝐹 = ( 𝐴 × 𝐵 ) )