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Metamath Proof Explorer

Theorem idinxpresid

Description: The intersection of the identity relation with the cartesian square of a class is the restriction of the identity relation to that class. (Contributed by FL, 2-Aug-2009) (Proof shortened by Peter Mazsa, 9-Sep-2022) (Proof shortened by BJ, 23-Dec-2023)

		Ref	Expression
	Assertion	idinxpresid	$⊢ I \cap (A \times A) = {I ↾}_{A}$

Proof

Step	Hyp	Ref	Expression
1		idinxpres	$⊢ I \cap (A \times A) = {I ↾}_{(A \cap A)}$
2		inidm	$⊢ A \cap A = A$
3	2	reseq2i	$⊢ {I ↾}_{(A \cap A)} = {I ↾}_{A}$
4	1 3	eqtri	$⊢ I \cap (A \times A) = {I ↾}_{A}$