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Metamath Proof Explorer

Theorem ssdmres

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

		Ref	Expression
	Assertion	ssdmres	$⊢ A \subseteq dom B \leftrightarrow dom ({B ↾}_{A}) = A$

Step	Hyp	Ref	Expression
1		dfss2	$⊢ A \subseteq dom B \leftrightarrow A \cap dom B = A$
2		dmres	$⊢ dom ({B ↾}_{A}) = A \cap dom B$
3	2	eqeq1i	$⊢ dom ({B ↾}_{A}) = A \leftrightarrow A \cap dom B = A$
4	1 3	bitr4i	$⊢ A \subseteq dom B \leftrightarrow dom ({B ↾}_{A}) = A$