Theorem fnmpo

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

	Ref	Expression
Hypothesis	fmpo.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
Assertion	fnmpo	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → 𝐹 Fn ( 𝐴 × 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fmpo.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
2		elex	⊢ ( 𝐶 ∈ 𝑉 → 𝐶 ∈ V )
3	2	2ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ V )
4	1	fmpo	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ V ↔ 𝐹 : ( 𝐴 × 𝐵 ) ⟶ V )
5		dffn2	⊢ ( 𝐹 Fn ( 𝐴 × 𝐵 ) ↔ 𝐹 : ( 𝐴 × 𝐵 ) ⟶ V )
6	4 5	bitr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ V ↔ 𝐹 Fn ( 𝐴 × 𝐵 ) )
7	3 6	sylib	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → 𝐹 Fn ( 𝐴 × 𝐵 ) )