Theorem ssdmres

Description: A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997)

	Ref	Expression
Assertion	ssdmres	⊢ ( 𝐴 ⊆ dom 𝐵 ↔ dom ( 𝐵 ↾ 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		df-ss	⊢ ( 𝐴 ⊆ dom 𝐵 ↔ ( 𝐴 ∩ dom 𝐵 ) = 𝐴 )
2		dmres	⊢ dom ( 𝐵 ↾ 𝐴 ) = ( 𝐴 ∩ dom 𝐵 )
3	2	eqeq1i	⊢ ( dom ( 𝐵 ↾ 𝐴 ) = 𝐴 ↔ ( 𝐴 ∩ dom 𝐵 ) = 𝐴 )
4	1 3	bitr4i	⊢ ( 𝐴 ⊆ dom 𝐵 ↔ dom ( 𝐵 ↾ 𝐴 ) = 𝐴 )