Theorem fnmpoi

Description: Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010)

	Ref	Expression
Hypotheses	fmpo.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
	fnmpoi.2	⊢ 𝐶 ∈ V
Assertion	fnmpoi	⊢ 𝐹 Fn ( 𝐴 × 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		fmpo.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
2		fnmpoi.2	⊢ 𝐶 ∈ V
3	2	rgen2w	⊢ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ V
4	1	fnmpo	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ V → 𝐹 Fn ( 𝐴 × 𝐵 ) )
5	3 4	ax-mp	⊢ 𝐹 Fn ( 𝐴 × 𝐵 )