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Metamath Proof Explorer

Theorem idinxpssinxp3

Description: Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 16-Mar-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	idinxpssinxp3	$⊢ I \cap (A \times A) \subseteq R \cap (A \times A) \leftrightarrow {I ↾}_{A} \subseteq R$

Proof

Step	Hyp	Ref	Expression
1		idinxpssinxp2	$⊢ I \cap (A \times A) \subseteq R \cap (A \times A) \leftrightarrow \forall x \in A x R x$
2		idrefALT	$⊢ {I ↾}_{A} \subseteq R \leftrightarrow \forall x \in A x R x$
3	1 2	bitr4i	$⊢ I \cap (A \times A) \subseteq R \cap (A \times A) \leftrightarrow {I ↾}_{A} \subseteq R$