Theorem mpoex

Description: If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by Mario Carneiro, 20-Dec-2013)

	Ref	Expression
Hypotheses	mpoex.1	⊢ 𝐴 ∈ V
	mpoex.2	⊢ 𝐵 ∈ V
Assertion	mpoex	⊢ ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ∈ V

Proof

Step	Hyp	Ref	Expression
1		mpoex.1	⊢ 𝐴 ∈ V
2		mpoex.2	⊢ 𝐵 ∈ V
3	2	rgenw	⊢ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V
4		eqid	⊢ ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
5	4	mpoexxg	⊢ ( ( 𝐴 ∈ V ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V ) → ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ∈ V )
6	1 3 5	mp2an	⊢ ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ∈ V