Theorem dmresi

Description: The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004)

	Ref	Expression
Assertion	dmresi	⊢ dom ( I ↾ 𝐴 ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		ssv	⊢ 𝐴 ⊆ V
2		dmi	⊢ dom I = V
3	1 2	sseqtrri	⊢ 𝐴 ⊆ dom I
4		ssdmres	⊢ ( 𝐴 ⊆ dom I ↔ dom ( I ↾ 𝐴 ) = 𝐴 )
5	3 4	mpbi	⊢ dom ( I ↾ 𝐴 ) = 𝐴