Theorem dmfex

Description: If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006) (Proof shortened by Andrew Salmon, 17-Sep-2011)

	Ref	Expression
Assertion	dmfex	⊢ ( ( 𝐹 ∈ 𝐶 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → 𝐴 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		fdm	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → dom 𝐹 = 𝐴 )
2		dmexg	⊢ ( 𝐹 ∈ 𝐶 → dom 𝐹 ∈ V )
3		eleq1	⊢ ( dom 𝐹 = 𝐴 → ( dom 𝐹 ∈ V ↔ 𝐴 ∈ V ) )
4	2 3	syl5ib	⊢ ( dom 𝐹 = 𝐴 → ( 𝐹 ∈ 𝐶 → 𝐴 ∈ V ) )
5	1 4	syl	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝐹 ∈ 𝐶 → 𝐴 ∈ V ) )
6	5	impcom	⊢ ( ( 𝐹 ∈ 𝐶 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → 𝐴 ∈ V )