Theorem dmresss

Description: The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021) Proof shortened and axiom usage reduced. (Proof shortened by AV, 15-May-2025)

	Ref	Expression
Assertion	dmresss	⊢ dom ( 𝐴 ↾ 𝐵 ) ⊆ dom 𝐴

Proof

Step	Hyp	Ref	Expression
1		resss	⊢ ( 𝐴 ↾ 𝐵 ) ⊆ 𝐴
2		dmss	⊢ ( ( 𝐴 ↾ 𝐵 ) ⊆ 𝐴 → dom ( 𝐴 ↾ 𝐵 ) ⊆ dom 𝐴 )
3	1 2	ax-mp	⊢ dom ( 𝐴 ↾ 𝐵 ) ⊆ dom 𝐴