Theorem ffdmd

Description: The domain of a function. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypothesis	ffdmd.1	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
Assertion	ffdmd	⊢ ( 𝜑 → 𝐹 : dom 𝐹 ⟶ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ffdmd.1	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
2		ffdm	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝐹 : dom 𝐹 ⟶ 𝐵 ∧ dom 𝐹 ⊆ 𝐴 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐹 : dom 𝐹 ⟶ 𝐵 ∧ dom 𝐹 ⊆ 𝐴 ) )
4	3	simpld	⊢ ( 𝜑 → 𝐹 : dom 𝐹 ⟶ 𝐵 )